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\title{\bf Partitioning sparse graphs into an independent set\\ and a forest of bounded degree}

\author{Fran\c{c}ois Dross \qquad Mickael Montassier \qquad Alexandre Pinlou \\
\small Universit\'e de Montpellier, CNRS, LIRMM, Montpellier, France \\
\small\tt\{francois.dross,mickael.montassier,alexandre.pinlou\}@lirmm.fr}

\date{\dateline{Feb 10, 2017}{Feb 7, 2018}{Mar 2, 2018}\\
\small Mathematics Subject Classifications: 05C10; 05C70; 05C15}

\begin{document}



\maketitle
\begin{abstract}
An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$.
We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition.
It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).
\end{abstract} 

\section{Introduction}

In this paper, unless we specify otherwise, all the graphs considered are simple graphs, without loops or multiple edges.

For $i$ classes of graphs ${\cal G}_1,\ldots, {\cal G}_i$, a
\emph{$({\cal G}_1,\ldots, {\cal G}_i)$-partition} of a graph $G$ is a partition of the vertices of $G$ into $i$ sets $V_1,\ldots,V_i$ such that, for all $1\le j \le i$, the graph $G[V_j]$ induced by $V_j$ belongs to ${\cal G}_j$. 

In the following we will consider the following classes of
graphs:
\begin{itemize}
\item ${\cal F}$ the class of forests,
\item ${\cal F}_d$ the class of forests with maximum degree at most
  $d$,
\item ${\Delta}_d$ the class of graphs with maximum degree at most $d$,
\item ${\cal I}$ the class of empty graphs (i.e. graphs with no edges).
\end{itemize}
For example, an $({\cal I},{\cal F},{\Delta}_2)$-partition of $G$ is a
vertex-partition into three sets $V_1,V_2,V_3$ such that $G[V_1]$ is
an empty graph, $G[V_2]$ is a forest, and $G[V_3]$ is a graph with maximum degree at most $2$.
Note that ${\Delta}_0 = {\cal F}_0 = I$ and $\Delta_1 = {\cal F}_1$.

The \emph{average degree} of a graph $G$ with $n$ vertices and $m$ edges, denoted by $\ad(G)$, is equal to $\frac{2m}{n}$. The \emph{maximum average degree} of a graph $G$, denoted by $\mad(G)$, is the maximum of $\ad(H)$ over all subgraphs $H$ of $G$. The \emph{girth} of a graph $G$ is the length of a smallest cycle in $G$, and infinity if $G$ has no cycle.

Many results on partitions of sparse graphs appear in the literature, where a graph is said to be sparse if it has a low maximum average degree, or if it is planar and has a large girth. The study of partitions of sparse graphs started with the Four Colour Theorem \cite{appel1,appel2}, which states that every planar graph admits an $({\cal I},{\cal I},{\cal I},{\cal I})$-partition. Borodin~\cite{Borodin} proved that every planar graph admits an $({\cal I}, {\cal F}, {\cal F})$-partition, and Borodin
and Glebov~\cite{borodin2001partition} proved that every planar graph
with girth at least $5$ admits an $({\cal I}, {\cal F})$-partition. Poh~\cite{Poh} proved that every planar graph admits an $({\cal F}_2,{\cal F}_2,{\cal F}_2)$-partition.


More recently, Borodin and Kostochka~\cite{borodin2014defective} showed that for all $j\ge 0$ and $k \ge 2j+2$, every graph $G$ with $\mad(G) < 2 \left(2-\frac{k+2}{(j+2)(k+1)} \right)$ admits a $({\Delta}_j,{\Delta}_k)$-partition. In particular, every graph $G$ with $\mad(G) < \frac{8}{3}$ admits an $({\cal I},{\Delta}_2)$-partition, and every graph $G$ with $\mad(G)<\frac{14}{5}$ admits an $({\cal I},{\Delta}_4)$-partition. With Euler's formula, this yields that planar graphs with girth at least $7$ admit $({\cal I},{\Delta}_4)$-partitions, and that planar graphs with girth at least $8$ admit $({\cal I},{\Delta}_2)$-partitions. Borodin and Kostochka~\cite{borodin2011vertex} proved that every graph $G$ with $\mad(G) < \frac{12}{5}$ admits an $({\cal I},{\Delta}_1)$-partition, which implies that that every planar graph with girth at least $12$ admits an $({\cal I},{\Delta}_1)$-partition.
This last result was improved by Kim, Kostochka and Zhu~\cite{kim2014improper}, who proved that every triangle-free graph with maximum average degree at most $\frac{11}{9}$ admits an $({\cal I},{\Delta}_1)$-partition, and thus that every planar graph with girth at least $11$ admits an $({\cal I},{\Delta}_1)$-partition.
In contrast with these results, Borodin, Ivanova, Montassier, Ochem and Raspaud~\cite{borodin2010vertex} proved that for every $d$, there exists a planar graph of girth $6$ that admits no $({\cal I},{\Delta}_d)$-partition. Montassier and Ochem~\cite{montassier2015near} showed that this implies that deciding if a planar graph of girth $6$ admits an $({\cal I},{\Delta}_d)$-partition in an NP-complete problem for all $d \ge 1$, and they proved that deciding if a planar graph of girth $7$ has an $({\cal I},{\Delta}_2)$-partition is NP-complete. Esperet, Montassier, Ochem, and Pinlou~\cite{esperet2013complexity} showed that deciding if a planar graph of girth $9$ has an $({\cal I},{\Delta}_1)$-partition is NP-complete.


It can be interesting to find partitions of sparse graphs into an independent set and a forest of bounded degree, that is $({\cal I},{\cal F}_d)$-partitions. Note that if a graph admits an $({\cal I},{\cal F}_d)$-partition, then it admits an $({\cal I},\Delta_d)$-partition, and that an $({\cal I},{\cal F}_1)$-partition is the same as an $({\cal I},\Delta_1)$-partition. Therefore the previous results imply that:
\begin{itemize}
\item for every $d$, there exists a planar graph of girth $6$ that admits no $({\cal I},{\cal F}_d)$-partition;
\item there exists a planar graph of girth at least $7$ that admits no $({\cal I},{\cal F}_2)$-partition;
\item there exists a planar graph of girth at least $9$ that admits no $({\cal I},{\cal F}_1)$-partition;
\item every planar graph with girth at least $11$ admits an $({\cal I},{\cal F}_1)$-partition.
\end{itemize}

Here are the main results of our paper:

\begin{theo} \label{t-main}
Let $M$ be a real number such that $M < 3$. Let $d \ge 0$ be an integer and let $G$ be a graph with $\mad(G) < M$. If $d \ge \frac{2}{3-M} - 2$, then $G$ admits an $({\cal I},{\cal F}_d)$-partition.
\end{theo}

\begin{theo} \label{t-main2}
Let $M$ be a real number such that $\frac{8}{3} \le M < 3$. Let $d \ge 0$ be an integer and let $G$ be a graph with $\mad(G) < M$.
If $d \ge \frac{1}{3-M}$, then $G$ admits an $({\cal I},{\cal F}_d)$-partition.
\end{theo}

By a direct application of Euler's formula, every planar graph with girth at least $g$ has maximum average degree less than $\frac{2g}{g-2}$. That yields the following corollary:

\begin{cor} \label{c-main}
Let $G$ be a planar graph with girth at least $g$.

\begin{enumerate}
\item If $g \ge 7$, then $G$ admits an $({\cal I},{\cal F}_5)$-partition. \label{c_1}

\item If $g \ge 8$, then $G$ admits an $({\cal I},{\cal F}_3)$-partition. \label{c_2}

\item If $g \ge 10$, then $G$ admits an $({\cal I},{\cal F}_2)$-partition. \label{c_3}
\end{enumerate}
\end{cor}

Corollaries~3.1 and~3.2 are obtained from Theorem~\ref{t-main2}, whereas Corollary~3.3 is obtained from Theorem~\ref{t-main}.
See Table~\ref{tab1} for an overview of the results on vertex partitions of planar graphs presented above.


\begin{table}
  \centering
  \begin{center}
    \begin{tabular}{|l|l|l|}
      \hline
      Classes & Vertex-partitions & References \\ \hline
      \multirow{3}{*}{Planar graphs} & $({\cal I},{\cal I},{\cal I},{\cal I})$ & The Four
                                                                                 Color Theorem \cite{appel1,appel2}\\
              & $({\cal I},{\cal F},{\cal F})$& Borodin \cite{Borodin} \\
              & $({\cal F}_2,{\cal F}_2,{\cal F}_2)$& Poh \cite{Poh} \\ \hline
      \multirow{1}{*}{Planar graphs with girth 5} & $({\cal I},{\cal F})$& Borodin and Glebov \cite{borodin2001partition} \\ \hline
      \multirow{1}{*}{Planar graphs with girth 6} & no $({\cal I},\Delta_d)$& Borodin et al. \cite{borodin2010vertex} \\ \hline
      \multirow{3}{*}{Planar graphs with girth 7} & no $({\cal I},\Delta_2)$& Montassier and Ochem~\cite{montassier2015near} \\ 
      & $({\cal I},\Delta_4)$& Borodin and Kostochka~\cite{borodin2014defective} \\ 
      & $({\cal I},{\cal F}_5)$ & Present paper \\ \hline
      \multirow{2}{*}{Planar graphs with girth 8} & $({\cal I},\Delta_2)$& Borodin and Kostochka~\cite{borodin2014defective} \\ 
      & $({\cal I},{\cal F}_3)$ & Present paper \\ \hline
      \multirow{1}{*}{Planar graphs with girth 9} & no $({\cal I},\Delta_1)$& Esperet et al.\cite{esperet2013complexity} \\  \hline
      \multirow{1}{*}{Planar graphs with girth 10} 
      & $({\cal I},{\cal F}_2)$ & Present paper \\ \hline
      \multirow{1}{*}{Planar graphs with girth 11} 
      & $({\cal I},\Delta_1)$ & Kim, Kostochka and Zhu~\cite{kim2014improper} \\ 
      \hline
    \end{tabular}
  \end{center}

  \caption{Known results on planar graphs.}
  \label{tab1}
\end{table}
\medskip

In term of tightness, it is not known whether every planar graph or girth $7$, $8$ or $10$ admits an $({\cal I},{\cal F}_d)$-partition for $d = 3$, $d = 2$ and $d = 1$ respectively. However, 
we note that Borodin, Ivanova, Montassier, Ochem and Raspaud~\cite{borodin2010vertex} constructed, for all $d$, a planar graph of girth $6$ with $16d + 14$ vertices that has no $({\cal I},{\Delta}_d)$-partition (and thus in particular no $({\cal I},{\cal F}_d)$-partition). Let us denote this graph by $G_d$. By Euler's formula, for every planar graph with girth at least $g$, $n$ vertices, $m$ edges, and at least one cycle, we have $\frac{m}{n-2} \le \frac{g}{g-2}$. Moreover, if a graph with $n$ vertices, $m$ edges has no cycle, then  $\frac{m}{n-2} \le 1 \le \frac{g}{g-2}$ for all $g \ge 3$.
Therefore for every planar graph of girth at least $6$ with $n$ vertices and $m$ edges, we have $\frac{2m}{n} \le 3\cdot\frac{n-2}{n}$. In particular, this is true for every subgraph of $G_d$. Thus, as $\frac{n-2}{n}$ increases when $n \ge 2$ increases, if we denote the number of vertices of $G_d$ by $n$, then $\mad(G_d) \le 3\cdot\frac{n-2}{n} = 3 - \frac{6}{n} = 3 - \frac{3}{8d+7} < 3 - \frac{3}{8d}$.
That shows the following claim.

\begin{claim}
For all integer $d$, there exists a graph with maximum average degree less than $M$, with $d =  \frac{3}{8(3-M)}$, that admits no $({\cal I},{\Delta}_d)$-partition (and thus no $({\cal I},{\cal F}_d)$-partition).
\end{claim}

That shows that Theorem~\ref{t-main2} is tight up to a multiplicative factor of $\frac{3}{8}$.

\section{Proof of Theorem~\ref{t-main} \label{p-main}}

Let $M < 3$, and let $d$ be an integer such that $d \ge \frac{2}{3-M} - 2$.
Let us call a \emph{good} $d$-partition of a graph $G$ a partition $(I,F)$ of the vertices of $G$ such that $I$ is an independent set of $G$, $G[F]$ is a graph with maximum degree at most $d$, and every cycle in $G[F]$ goes through a vertex with degree $2$ in $G$. Note that for any graph $G$, if $G$ admits a good $d$-partition, then $G$ admits an $({\cal I},{\cal F}_d)$-partition: while there is a vertex $v$ with degree $2$ in $G$ that is in $F$ and has two neighbours in $F$, move $v$ from $F$ to $I$.
Theorem~\ref{t-main} is implied by the following lemma:
\begin{lemm} \label{l-main}
Every graph $G$ with $\mad(G) < M$ has a good $d$-partition.
\end{lemm}



Our proof uses the discharging method. For the sake of contradiction, assume that Lemma~\ref{l-main} is false. Let $G$ be a counterexample to Lemma~\ref{l-main} with minimum order.

For all $k$, a vertex of degree $k$, at least $k$, or at most $k$ in $G$ is a \emph{$k$-vertex}, a \emph{$k^+$-vertex}, or a \emph{$k^-$-vertex} respectively. A $(d+1)^-$-vertex is a \emph{small} vertex, and a $(d+2)^+$-vertex is a \emph{big} vertex. Let $v$ be a vertex of $G$ and $w$ be a neighbour of $v$.
For all $k$, if $w$ has degree $k$, at least $k$, or at most $k$ in $G$, then $w$ is a \emph{$k$-neighbour}, a \emph{$k^+$-neighbour}, or a \emph{$k^-$-neighbour} of $v$ respectively. A neighbour of $v$ that is a big vertex is a \emph{big neighbour} of $v$, and a neighbour of $v$ that is a small vertex is a \emph{small neighbour} of $v$. We start by proving some lemmas on the structure of $G$. Specifically, we prove that some configurations are reducible, and thus cannot occur in $G$.

\begin{lemm} \label{1v}
There are no $1^-$-vertices in $G$.
\end{lemm}

\begin{proof}
Assume there is a $1^-$-vertex $v$ in $G$. The graph $G-v$ has one fewer vertex than $G$, and thus, by minimality of $G$, admits a good $d$-partition $(I,F)$. If $v$ has no neighbours in $I$, then we can add it to $I$. Otherwise, it has no neighbours in $F$, and we can add it to $F$. In both cases, that leads to a good $d$-partition of $G$, a contradiction.
\end{proof}


\begin{lemm} \label{2v}
Every $2$-vertex has at least one big neighbour.
\end{lemm}

\begin{proof}
Assume $v$ is a $2$-vertex adjacent to two small vertices, $u$ and $w$. The graph $G-v$ has one fewer vertex than $G$, and thus, by minimality of $G$, admits a good $d$-partition $(I,F)$. If $u$ and $w$ are both in $F$, then we can put $v$ in $I$, and if they are both in $I$, then we can put $v$ in $F$. Therefore without loss of generality, we can assume that $u \in I$ and $w \in F$. If $w$ has no neighbours in $I$, then we can put it in $I$, and put $v$ in $F$. Therefore we can assume that $w$ has at least one neighbour in $I$ and thus at most $d-1$ neighbours in $F$ (since $w$ is a small vertex in $G$). Then $v$ has at most one neighbour in $F$, and this neighbour $w$ has at most $d-1$ neighbours in $G[F]$, thus we can add $v$ to $F$.
In every case, this leads to a good $d$-partition of $G$, a contradiction.
\end{proof}

A $2$-vertex is a \emph{light $2$-vertex} if it is adjacent to a small vertex, and it is a \emph{heavy $2$-vertex} otherwise. Note that by Lemma~\ref{2v}, each $2$-vertex has at most one small neighbour. 

\begin{lemm} \label{nobud}
Let $B$ be a set of small $3^+$-vertices such that $G[B]$ is a tree. There exists a $3^+$-vertex $v \notin B$ that is adjacent to a vertex of $B$.
\end{lemm}

\begin{proof}
Assume that the lemma is false, that is every vertex that is not in $B$ but has a neighbour in $B$ is a $2$-vertex. By minimality of $G$, $G - B$ admits a good $d$-partition $(I,F)$. For every vertex $v$ in $B$, successively, we put $v$ in $I$ if $v$ has no neighbours in $I$ and we put it in $F$ otherwise. Note that this way a vertex that we add to $F$ has at most $d$ neighbours that are not in $I$, and we cannot make any cycle in $G[F]$ that does not go through a $2$-vertex, since $G[B]$ is a tree. Thus we have a good $d$-partition of $G$, a contradiction.
\end{proof}

Let $B$ be a set of small $3^+$-vertices such that:
\begin{enumerate}[label=(\alph*)]
\item $G[B]$ is a tree, 
\item there is only one edge that links a vertex of $B$ to a $3^+$-vertex $u$ outside of $B$,
\item $u$ is a big vertex. 
\end{enumerate}
We call $B$ a \emph{bud} with father $u$. Note that every vertex that has a neighbour in a bud is a $2$-vertex or a big vertex, or is in that bud. Therefore two different buds always have an empty intersection.

Let us build a structure by the following three steps. We note that some choices can be made in the construction, specifically the order in which the vertices are considered, and thus that the construction may not be unique. However, we do not care about that, and just build one such structure. We call this structure the \emph{light forest} $L$.

We start with $L = (\emptyset, \emptyset)$, the graph with no vertices and no edges.
\begin{enumerate}
\item While there are light $2$-vertices that are not in $L$, do the following. Pick a light $2$-vertex $v$, and let $u$ be the big neighbour of $v$ (that exists by Lemma~\ref{2v}). Add to $L$ the vertex $v$, the edge $uv$, and the vertex $u$ (if it is not already in $L$). Also set that $u$ is the \emph{father} of $v$ (and $v$ is a \emph{son} of $u$). See Figure~\ref{figL}, left. Note that by doing this, we obtain a star forest with only big vertices and light $2$-vertices. Also note that the set of the big vertices and the set of the light $2$-vertices are independent sets in $L$ (but not necessarily in $G$).

\item While there are buds whose vertices are not all included in $L$, do the following. Pick a bud $B$. Let $u$ be the father of $B$, and let $v$ be the vertex of $B$ adjacent to $u$. Add $G[B]$ to $L$, as well as the edge $uv$, and the vertex $u$ (if it is not already in $L$). The vertex $u$ is the father of $v$, and the father/son relationship in $B$ is that of the tree $G[B]$ rooted at $v$. See Figure~\ref{figL}, middle. We recall that the buds do not share vertices. Since we add the vertices of the buds to $L$ all in one go, in the end all the vertices of the buds are in $L$.
 Note that each iteration of this step always adds to $L$ a tree with at most one vertex ($u$) that was already in $L$. Hence, $L$ is still a forest. Moreover, it is still a rooted forest since the tree we add has the orientation of a tree rooted at $u$.

\item While, for some $k$, there exists a big $k$-vertex $w \in L$ that has $k-1$ sons in $L$ and a $2$-neighbour $v$ that is not in $L$, do the following. Let $u$ be the neighbour of $v$ distinct from $w$. Note that $v$ is a heavy $2$-vertex (since it was not added to $L$ in Step 1), therefore $u$ is a big vertex. Add to $L$ the vertex $v$, the edges $uv$ and $vw$, and the vertex $u$ (if it is not already in $L$). We set that $v$ is the father of $w$, and that $u$ is the father of $v$. See Figure~\ref{figL}, right. Note that this operation just takes a root of $L$, adds a vertex as a father of this root and another vertex as a father of that new vertex. Therefore, $L$ is still a rooted forest. Also note that each of the set of the big vertices and the set of the $2$-vertices remains independent in $L$. 
\end{enumerate}


\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.95]
\coordinate (v) at (-10.5,0);
\coordinate (u) at (-12,0);
\coordinate (v') at (-9,0);


\draw (v) node [below] {$v$} ; 
\draw (-12.05,0) node [left] {$u$} ; 

\draw (u) -- (-11.25,0);
\draw [>=stealth,<-] (-11.25,0) -- (v);
\draw (v) -- (v');

\draw [fill=black](v) circle (1.5pt) ;
\draw [fill=white](v') circle (1.5pt) ;
\draw [fill=white](u) circle (4pt) ;


\coordinate (u) at (-8,0);
\coordinate (v) at (-6.5,0);
\coordinate (v1) at (-5,1);
\coordinate (v2) at (-5,0);
\coordinate (v3) at (-5,-1);
\coordinate (v11) at (-4,1.25);
\coordinate (v12) at (-4,0.75);


\draw (v) node [below] {$v$} ; 
\draw (-8.05,0) node [left] {$u$} ; 

\draw (u) -- (-7.25,0);
\draw [>=stealth,<-] (-7.25,0) -- (v);
\draw (v) -- (-5.75,0.5);
\draw [>=stealth,<-] (-5.75,0.5) -- (v1);
\draw (v) -- (v2);
\draw (v2) -- (-4.25,0);
\draw (v) -- (v3);
\draw (v3) -- (-4.25,-1);
\draw (v1) -- (v11);
\draw (v11) -- (-3.5,1.25);
\draw (v1) -- (v12);
\draw (v12) -- (-3.5,0.75);

\draw [fill=black](v) circle (1.5pt) ;
\draw [fill=black](v1) circle (1.5pt) ;
\draw [fill=black](v2) circle (1.5pt) ;
\draw [fill=black](v3) circle (1.5pt) ;
\draw [fill=black](v11) circle (1.5pt) ;
\draw [fill=black](v12) circle (1.5pt) ;
\draw [fill=white](u) circle (4pt) ;


\coordinate (v1) at (0,0);
\coordinate (u1) at (-2,0);
\coordinate (w1) at (2,0); 

\draw (v1) node [below] {$v$} ;
\draw (w1) node [below left] {$w$} ;
\draw (-2.05,0) node [left] {$u$} ;

\draw (u1) -- (-1,0);
\draw [>=stealth,<-] (-1,0) -- (v1);
\draw (v1) -- (1,0);
\draw [>=stealth,<-] (1,0) -- (w1);
\draw (w1) -- (3,0);
\draw [>=stealth,<-] [dashed] (3,0) -- (3.5,0);
\draw (w1) -- (2.71,0.71);
\draw [>=stealth,<-] [dashed] (2.71,0.71) -- (3.06,1.06);
\draw (w1) -- (2,1);
\draw [>=stealth,<-] [dashed] (2,1) -- (2,1.5);
\draw (w1) -- (2.71,-0.71);
\draw [>=stealth,<-] [dashed] (2.71,-0.71) -- (3.06,-1.06);
\draw [<-] (w1) -- (2,-1);
\draw [>=stealth,<-] [dashed] (2,-1) -- (2,-1.5);

\draw [fill=black](v1) circle (1.5pt) ;
\draw [fill=white](u1) circle (4pt) ;
\draw [fill=black](w1) circle (4pt) ;

\end{tikzpicture}
\end{center}
\caption{The construction of the light forest $L$. The big vertices are represented with big circles, and the small vertices with small circles. The filled circles represent vertices whose incident edges are all represented. The dashed lines are the continuation of the light forest. The arrows point from son to father in $L$.\label{figL}}
\end{figure}

As noticed previously, $L$ is a rooted forest. We say that a vertex $v$ is a \emph{descendant} of a vertex $u \ne v$ in $L$ if there are vertices $v_0 = v$, $v_1$, \ldots, $v_k = u$ in $L$, such that for $i \in \{0,1,\ldots,k-1\}$, $v_{i+1}$
is the father of $v_i$ in $L$.
Consider a vertex $v$ in $L$. If $v$ is a heavy $2$-vertex, then $v$ was added in Step 3, and its two incident edges were added at the same time. If $v$ is a big vertex and is not the root of its connected component in $L$, then the father of $v$ was added in Step $3$, thus all the neighbours of $v$ distinct from its father are its sons in $L$. If $v$ is a small $3^+$-vertex in $L$, then $v$ was added in Step $2$ and thus is in a bud.
Therefore, a vertex $v$ in $L$ is incident to an edge that is not in $L$ only if either $v$ is a big vertex and the root of its connected component in $L$, or $v$ is a light $2$-vertex, or $v$ is in a bud. The \emph{pending vertices} of $L$ are the vertices that are not in $L$ but are adjacent to a light $2$-vertex. Note that the pending vertices are small.

Let $B$ be a bud with father $u$. Let $S \subseteq V(G) \setminus
(B\cup\{u\})$
and let $(I,F)$ be a good $d$-partition of $S\cup \{u\}$ such that $u$
either is in $I$ or has at most $d-1$ neighbours in $F$. We show that
we can extend the good $d$-partition to $S \cup \{u\} \cup B$. We proceed as
follows: for every vertex $v\in B$ successively, we add $v$ to $I$ if it has no
neighbours in $I$ or to $F$ otherwise. The vertices in $I$ clearly
form an independent set. Moreover, $G[F]$ has maximum degree at most
$d$ and every cycle of $G[F]$ goes through a $2$-vertex by
construction of a bud. This leads to a good $d$-partition of $S\cup B \cup
\{u\}$. We call that
process \emph{colouring the bud $B$}.


Let $v$ be a $2$-vertex of $L$, $u$ its father and $D_v$ the set of the
descendants of
$v$. Let $(I,F)$ be a good $d$-partition of $S \subseteq V(G) \setminus (D_v
\cup \{u,v\})$. We show that we can extend the good $d$-partition to $S \cup
D_v\cup
\{v\}$. We proceed as follows:
\begin{enumerate}[align=left]
\item[Step $1$.] We add every big vertex of $D_v$ to $I$. We can do this, since big vertices
  form an independent set in $L$ by construction, and as we noted previously, big vertices that are in $L$ and are not the root of their component (in particular big vertices that are in $D_v$) have no incident edge outside of $L$.
\item[Step $2$.]\label{item:pending} While there is a pending vertex $w\in S$ that has no
neighbours in $I$, we add one such vertex to $I$.
\item[Step $3$.] We add every $2$-vertex of $D_v$ and $v$ to $F$. We can do this, since the
  $2$-vertices of $D_v$ form a stable set in $L$. Moreover,
  Step~\ref{item:pending} ensures that the maximum degree of $G[F]$ is
  at most $d$.
\item[Step $4$.] Finally, we colour every bud. Indeed, the father of every bud whose vertices are in
  $D_v$ has been put in $I$ in Step $1$.
\end{enumerate}
This leads to a good $d$-partition of $S\cup D_v \cup \{v\}$. We call that
process \emph{descending $v$}.

\begin{lemm} \label{kroot}
For all $k$, there are no big $k$-vertices in $G$ that are in $L$ and have $k$ sons in $L$.
\end{lemm}

\begin{proof}
Let $u$ be a big $k$-vertex that has $k$ sons in $L$. Note that this implies that $u$ is the root of its connected component in $L$. Let $C$ be the connected component of $u$ in $L$. Let $H = G-V(C)$. The graph $H$ has fewer vertices than $G$ and thus, by minimality of $G$, $H$ admits a good $d$-partition $(I,F)$. Let $N$ be the set of the $2$-neighbours of $u$. We descend every vertex of $N$. Note that this implies that every vertex of $N$ is put in $F$. We add $u$ to $I$. Then we colour every bud of father $u$. This leads to a good $d$-partition of $G$, a contradiction.
\end{proof}

\subsection*{Discharging procedure}
Let $\epsilon = 3 - M$. 
Recall that $d \ge \frac{2}{3-M} - 2 = \frac{2}{\epsilon}-2$, therefore
 $\epsilon \ge \frac{2}{d+2} > 0$. 
For all $k$, we assign to each $k$-vertex a charge equal to $k - M = k - 3 + \epsilon$. Note that since $M$ is bigger than the average degree of $G$, the sum of the charges of the vertices is negative. The initial charge of each $3^+$-vertex is at least $\epsilon$, and thus is positive.

For every big vertex $v$, $v$ gives charge $1 - \epsilon$ to each of its $2$-neighbours that are its sons in $L$, does not give anything to its father in $L$ (if it has one), and gives $\frac{1 - \epsilon}{2}$ to its other $2$-neighbours.

\begin{lemm}\label{nneg}
Every vertex has non-negative charge at the end of the procedure.
\end{lemm} 

\begin{proof}
The small $3^+$-vertices start with a non-negative charge, and do not give or receive charge throughout the procedure, thus they have non-negative charge at the end of the procedure. 

Every $2$-vertex is either in $L$, in which case it receives $1 - \epsilon$ from its father in $L$, or is not in $L$ and is a heavy $2$-vertex, in which case it receives $\frac{1 - \epsilon}{2}$ from each of its neighbours. As $2$-vertices have charge $\epsilon - 1$ at the start of the procedure, and as they receive $1 - \epsilon$, they have charge $0$ at the end of the procedure.

Let $v$ be a big $k$-vertex. By Lemma~\ref{kroot}, $v$ has at most $k-1$ sons in $L$. Moreover, by construction of $L$, if $v$ has  $k-1$ sons in $L$, then either its neighbour that is not its son in $L$ is a $3^+$-vertex, or it is the father of $v$ in $L$ (and in both cases $v$ does not give charge to this vertex).
Therefore $v$ gives charge amounting to at most $(k-1)(1 - \epsilon)$. Since its initial charge is $k - 3 + \epsilon$, in the end it has at least $k - 3 + \epsilon - (k-1)(1 - \epsilon)= k\epsilon - 2$. Since every big vertex has degree at least $d + 2 \ge \frac{2}{\epsilon}$, the final charge of each big vertex is at least $\frac{2}{\epsilon}\epsilon - 2 = 2 - 2 = 0$.
\end{proof}

By Lemma~\ref{nneg}, every vertex has non-negative charge at the end of the procedure, thus the sum of the charges at the end of the procedure is non-negative. Since no charge was created nor removed, this is a contradiction with the fact that the initial sum of the charges is negative. That ends the proof of Lemma~\ref{l-main}.





\section{Proof of Theorem~\ref{t-main2}}

This proof is similar to the proof of Theorem~\ref{t-main} above.
Let $\frac{8}{3} \le M < 3$, and let $d$ be an integer such that $d \ge \frac{1}{3-M}$.
We define good $d$-partitions as in Section~\ref{p-main}.
Theorem~\ref{t-main2} is implied by the following lemma:
\begin{lemm} \label{l-main2}
Every graph $G$ with $\mad(G) < M$ has a good $d$-partition.
\end{lemm}

For the sake of contradiction, assume that Lemma~\ref{l-main2} is false.  Let $G$ be a counterexample to Lemma~\ref{l-main2} with minimum order.

We take the same definitions as before. Lemmas \ref{1v}--\ref{kroot} of the previous section still hold in this setting.
Frank and Gy{\'a}rf{\'a}s~\cite{gyarfas} prove the following theorem:

\begin{theo}[Frank and Gy{\'a}rf{\'a}s~\cite{gyarfas}] \label{gya}
Let $H = (V,E)$ be a graph, and let $\omega : V \rightarrow \mathbb{N}$.
There exists an orientation such that $\forall v \in V, d^+(v) \ge \omega(v)$ if and only if for all $X \subset V$, $\omega(X) \le |\{uv \in E, u \in X\}|$.
\end{theo}

Given $H = (V,E)$ and $\omega : V \rightarrow \mathbb{N}$, a \emph{good $\omega$-orientation} of $H$ is an orientation of $H$ such that $\forall v\in V, d^+(v) \ge \omega(v)$.
We prove some additional lemmas. 

\begin{lemm} \label{omega}
Let $H = (V,E)$ be a graph on $n \ge 1$ vertices and $m$ edges. Let $\omega : V \rightarrow \mathbb{N}$ such that $\omega(V) \le m$.
There exists a subgraph $S$ of $H$ with at least one vertex such that $S$ admits a good $\omega$-orientation.
\end{lemm}

\begin{proof}
For a graph $I$ and a set $X \subset V(I)$, let $e_I(X) = |\{uv \in E(I), u \in X\}|$. If $\omega(X) \le e_I(X)$, we say that $X$ is \emph{good} in $I$.

If every subset of $V$ is good in $H$, then by Theorem~\ref{gya}, we have a good $\omega$-orientation of $H$. Therefore we may assume that there is a subset of $V$ that is not good in $H$. Let $X$ be a maximum subset of $V$ that is not good in $H$. Let $Y = V - X$, and let $H' = H[Y]$. Note that $V$ is good in $H$, since $\omega(V) \le m$, so $Y \ne \emptyset$.

If every subset of $Y$ is good in $H'$, then by Theorem~\ref{gya}, we have a good $\omega$-orientation of $H'$. Therefore there is a $Z \subset Y$ such that $Z$ is not good in $H'$, i.e. $\omega(Z) > e_{H'}(Z)$. As $X$ is not good in $H$, we also have $\omega(X) > e_H(X)$. Therefore we have $\omega(X \cup Z) = \omega(X) + \omega(Z) > e_H(X) + e_{H'}(Z) = |\{uv \in E(H), u \in X\}| + |\{uv \in E(H'), u \in Z\}| = |\{uv \in E(H), u \in (X \cup Z)\}| = e_H(X \cup Z)$. Therefore $X \cup Z$ is not good in $H$, which contradicts the maximality of $X$.
%Let us assume by contradiction that the statement of the lemma is false, and let us consider a weighted graph $H$ that has no good $\omega$-orientation, with $n$ vertices and $m$ edges, such that $n+m$ is minimized.
%
%If every vertex $u$ of $H$ has degree at least $2\omega(u)$, then we do the following: let $\widehat H$ be the graph $H$, where we add edges between odd degree vertices until every vertex has even degree (note that we can do this since the number of odd degree vertices is always even). The graph $\widehat H$ is Eulerian, therefore there is a cycle that goes exactly once through each edge. Orienting this cycle leads to an orientation of $\widehat H$ where every vertex of degree $2k$ has exactly $k$ incoming edges and $k$ outgoing edges. Let $\sigma$ be the restriction of this orientation to $H$. For every even-degree vertex $u$, $u$ has degree at least $2\omega(u)$ in $H$ and in $\widehat H$, thus it has out-degree at least $\omega(u)$ in $\widehat H$, and thus it has out-degree at least $\omega(u)$ in $H$. For every odd-degree vertex $u$, $u$ has degree at least $2\omega(u)+1$ in $H$ and at least $2\omega(u)+2$ in $\widehat H$, thus it has out-degree at least $\omega(u)+1$ in $\widehat H$, and thus it has out-degree at least $\omega(u)$ in $H$. Therefore $\sigma$ is a good $\omega$-orientation of $H$, a contradiction.
%
%Therefore we can assume that $H$ has a vertex $u$ that has degree less than $2\omega(u)$. The vertex $u$ is not the only vertex in the graph (since $m \ge \sum_{v \in V(H)} \omega(v)$). If $u$ has degree at most $\omega(u)$, then by minimality of $H$, $H - u$ admits a good $\omega$-orientation, a contradiction. So the degree of $u$ is at least $\omega(u)+1$. Let $H'$ be the graph $H$, where only $2(d(u) - \omega(u))$ of the edges of $u$ are kept (i.e. $2\omega(u) - d(u)$ edges of $u$ are removed), and $\omega(u)$ is replaced by $d(u) - \omega(u)$. Let us call $\omega'$ the new weighting of the vertices of $H'$. Note that $\sum_{v \in V(H')} \omega'(v) = \sum_{v \in V(H)} \omega(v) +d(u) - 2\omega(u) \le m + d(u) - 2\omega(u) = |E(H')|$. Therefore, by minimality of $H$, there exists a subgraph $S$ of $H'$ that admits a good $\omega'$-orientation. If $u \notin S$, then $\omega$ and $\omega'$ coincide on $V(S)$, so $S$ admits a good $\omega$-orientation. If $u \in S$, then we can orient the $2\omega(u) - d(u)$ edges of $u$ in $H$ that are not oriented (because they are not in $H'$) away from $u$, and in this orientation the outdegree of $u$ is at least $\omega'(u) + 2\omega(u) - d(u) = d(u) - \omega(u) + 2\omega(u) - d(u) = \omega(u)$. This leads to a good $\omega$-orientation of $S$. In both cases, $S$ is a subgraph of $H$ with a good $\omega$-orientation, a contradiction.
\end{proof}


We recall that $L$ is the light forest of $G$.
\begin{lemm} \label{graphH1}
Let $U$ be a non-empty subset of $V(L)$ with no small $3^+$-vertices. Let
$H=G[U]$ (i.e. the subgraph of $G$ induced by the 2-vertices and the
big vertices of $U\subseteq L$). Suppose:
\begin{enumerate}
\item There is an orientation of the edges of $H$ such that every
  2-vertex in $H$ has at least one out-going edge, and for all $i\ge
  1$, every big $(d+i+1)$-vertex in $G$  has at least $i$ out-going
  edges.
\item There are no $1^-$-vertices in $H$.
\end{enumerate}
Then $H$ contains an edge that is not in $L$ and that is incident to a
big vertex.
\end{lemm}

The graph $H$ of Lemma \ref{graphH1} is as follows: it is composed
by subtrees of $L$ plus some additional edges (that do not belong to
$L$). Such edges are edges between light $2$-vertices, and maybe
edges between roots of trees of $L$. The aim of Lemma \ref{graphH1}
is to prove the existence of such latter edges.

The orientation in Lemma \ref{graphH1} does not correspond to the
orientation defined by the father/son relation. This orientation will
allow us to extend a partial partition $(I,F)$: consider a big
$(d+i+1)$-vertex $v$ being in $F$. Vertex $v$ must have at least $i+1$
neighbours in $I$. The orientation will point towards $i$ sons of $v$ that
will be added to $I$. Moreover we will see that $v$ will have one
extra neighbour in $I$: either its father in $L$, or a neighbour outside $L$.

\bigskip
\begin{proof}[Proof of Lemma~\ref{graphH1}]
Assume the lemma is false: every edge of $H$ that is not in $L$ is
between two 2-vertices. Let $R_0$ be the set of the vertices of $H$ that
are not the descendants in $L$ of a vertex of $H$. In particular, $R_0$
contains the roots of $L$ that are in $H$, plus big vertices that have no ancestor in $U$. Note that $R_0$ contains only big
vertices ; otherwise, $H$ would contain $1$-vertices. Moreover, $H - R_0$ has at least one vertex, otherwise $U$ would contain only big vertices, there would be an edge between two big vertices, and this edge could not be in $L$. Let $S$ be the set
of the vertices that are not in $H$, but are descendants of vertices
of $H$.


By minimality of $G$, the graph $G - (V(H-R_0) \cup S)$ admits a good
$d$-partition $(I,F)$. While there is a vertex $v\in R_0$ that is in $F$ and
has no neighbours in $I$, we put $v$ in $I$. Now we can assume that
every vertex in $R_0\cap F$ has a neighbour in $I$. Let $R=R_0$ (in
the following we describe a procedure that modifies $R$ but we need to
refer to vertices of $R_0$). While there is
a vertex in $R$, do the following:

\begin{itemize}
\item Suppose $u$ is in $I$. We descend every 2-vertex with father $u$
  (by the procedure every 2-vertex is added to $F$) and colour every
  bud with father $u$. This leads to a good $d$-partition of $u$ and all
  its descendants.
\item Suppose $u$ is in $F$. We remove $u$ from $R$. For every 2-vertex $v$ in $H$ with father
  $u$ such that the edge $uv$ is oriented from $u$ to $v$ (according to the orientation defined in the statement of the lemma), we
  first add $v$ to $I$, and then add the son of $v$ to $F$ and
  $R$. By hypothesis, if $u$ is a $(d+i+1)$-vertex, then it has at
  least $i$ outgoing edges. These edges lead to sons of $u$:
  \begin{itemize}
  \item either $u\in R_0$, thus $u$ has no ancestors in $H$ by
    construction and all its neighbours in $H$ are its sons ; moreover
    it has a neighbour outside $H$ that is in $I$ (by construction).
  \item or, $u \in R \setminus R_0$, this means that $u$ was added to
    $R$ during the procedure, this implies that his father, say $w$, is
    a 2-vertex added to $I$ and the edge $wv$ is oriented from $w$ to
    $v$ (as every 2-vertex has an out-going edge by hypothesis). It
    follows that all the out-going neighbours of $u$ are sons of $u$.
  \end{itemize}
  It follows that $u$ has at least $i+1$ neighbours in $I$, and so all
  other neighbours can be added to $F$ without violating the degree
  condition on $F$. Now we
  descend every 2-vertex $v\notin H$ with father $u$, and every
  $2$-vertex $v\in H$ with father $u$ such that the edge $uv$ is
  oriented from $v$ to $u$, and colour every bud with father $u$. The
  only problem that could occur is when two adjacent light $2$-vertices $\ell$ and
  $\ell'$ are added to $I$: in that case, since $\ell$ and $\ell'$
  were added to $I$, the edge that links $\ell$ (resp. $\ell'$) to its
  father is towards $\ell$ (resp. $\ell'$); it follows that one of
  $\ell,\ell'$ has no out-going edges, contradicting the hypothesis.
\end{itemize}
In all cases, that leads to a good $d$-partition of $G$, a contradiction.
\end{proof}


\begin{lemm} \label{graphH2}
Let $U$ be a non-empty subset of $V(L)$ with no small $3^+$-vertices. Let
$H=G[U]$ (i.e. the subgraph of $G$ induced by the 2-vertices and the
big vertices of $U\subseteq L$). Suppose that $H$ has no edge linking
two roots of two connected components of $L$. Let us denote by
$n_2^G(H)$ the number of vertices of $H$ that are 2-vertices in
$G$. Then,
$$|E(H)| < n_2^G(H) + \sum_{big\ v\in H} \left( d_G(v) -d -1 \right).$$
\end{lemm}

\begin{proof}
  By contradiction, suppose there exists such a subgraph $H$ with
  $|E(H)| \ge n_2^G(H) + \sum_{big\ v\in H} \left( d_G(v) -d -1
  \right)$. Let us define a weight function $\omega:V(H)\to
  \mathbb{N}$ such that, for every 2-vertex $u$, $\omega(u)=1$ and,
  for every $(d+i+1)$-vertex $v$ with $i \ge 1$ in $\mathbb{N}$, $\omega(v)=i$. By hypothesis,
  $|E(H)|\ge \sum_{v\in V(H)} \omega(v)$. By Lemma \ref{omega}, $H$ contains a non empty
  subgraph $I$ that has a good $\omega$-orientation.

  Suppose that $I$ has a vertex of degree 1, say $v$, and let $u$ be
  the neighbour of $v$ in $I$. As $\omega(v)\ge 1$, the only edge
  incident to $v$ goes from $v$ to $u$. It follows that, for all
  $w\neq v$, $w$ has the same number of outgoing edges in $I$ and in
  $I-\{v\}$. Hence $I-\{v\}$ is a subgraph of $H$ with at least one
  vertex (it contains $u$) and has a good $\omega$-orientation. By
  successively removing vertices of degree 1 from $I$, we can assume
  that $I$ has no $1$-vertices.

  By Lemma \ref{graphH1}, $I$ has an edge $e$ that is not in $L$ and is incident
  to a big vertex. As no light $2$-vertex is adjacent to a big vertex
  besides its father, edge $e$ has to link the roots of two connected
  components of $L$, contradicting the hypothesis.
\end{proof}


%
%\begin{lemm} \label{graphH1}
%Let $H$ be a subgraph of $G$ whose set of vertices is a non-empty subset of $V(L)$ with no small $3^+$-vertices. Assume that there is an orientation of the edges of $H$ such that every $2$-vertex in $H$ has at least one out-going edge, and for all $i\ge 1$, every big $d+i+1$-vertex has at least $i$ out-going edges. Also assume that there is no vertex of degree $1$ in $H$. Then $H$ contains an edge that is not in $L$ and that is incident to a big vertex.
%\end{lemm}
%
%\begin{proof}
%Assume the lemma is false, that is that every edge of $H$ that is not in $L$ is between two $2$-vertices.
%Let $R$ be the set of the roots of $L$ that are in $H$, plus the vertices of $H$ that are not the descendants of a vertex in $H$. Note that $R$ contains only big vertices, therefore if there is no vertex outside of $R$, then there is an edge between two big vertices, and that edge cannot be in $L$. We may thus assume that $V(H-R) \ne \emptyset$. Let $S$ be the set of the vertices that are not in $H$, but are descendants of vertices of $H$.
%
%By minimality of $G$, the graph $G - V(H-R) - S$ admits a good $d$-partition $(I,F)$. While there is a vertex $v \in R$ that is in $F$ and has no neighbour in $I$, we put $v$ in $I$. Now we can assume that every vertex in $R \cap F$ has a neighbour in $I$. 
%
%While there is a vertex in $R$, do what follows:
%\begin{itemize}
%\item Pick a vertex $u \in R$.
%
%\item Suppose $u$ is in $I$. We descend every $2$-vertex with father $u$ and add every $2$-vertex with father $u$ to $F$.
%
%\item Suppose $u$ is in $F$. We colour every bud with father $u$. For every $2$-vertex $v \notin H$ with father $u$, we descend $v$ and add $v$ to $F$.
%For every $2$-vertex $v \in H$ with father $u$ such that the edge $uv$ is oriented from $v$ to $u$, descend $v$ and add $v$ to $F$. For every $2$-vertex $v \in H$ with father $u$ such that the edge $uv$ is oriented from $u$ to $v$, add $v$ to $I$, then add the son of $v$ to $F$ and to $R$ if $v$ has a son. Note that if $u$ is a $(d+i+1)$-vertex, then $u$ has at least $i$ out-going edges, and thus at least $i$ sons in $I$, and its father is in $I$, so it has at most $d$ neighbours that are not in $I$. The only problem that could occur is if we have two light $2$-vertices $l$ and $l'$ of $L$ that are adjacent in $G$ and that are put in $I$. Since $l$ and $l'$ are put in $I$, the edge that links $l$ to its father is towards $l$, and the edge that links $l'$ to its father is towards $l'$. Now one of $l$ or $l'$ has no out-going edge, a contradiction.
%\end{itemize}
%
%Note that throughout the procedure, every vertex that is not in $F \cup I$ is the descendent of a vertex in $R$. Therefore this leads to a good $d$-partition of $G$, a contradiction.
%\end{proof}
%
%\begin{lemm} \label{graphH2}
%Let $H$ be a subgraph of $G$ whose set of vertices is a subset of $V(L)$ with no small $3^+$-vertices, such that $H$ has no edge that links the roots of two connected components of $L$. Then in $H$, the total number of edges is less than the number of $2$-vertices that are in $H$, plus $\sum_{v \text{ big } \in H} d_G(v)-d-1$. 
%\end{lemm}
%
%\begin{proof}
%Let us assume that the lemma is false, and let us consider a counter-example $H$.
%Let us define a weight function $\omega$ on the vertices of $H$: for every $2$-vertex $u$ of $H$, $\omega(u) = 1$, and for every $d+i+1$-vertex $v$, $\omega(v) = i$. Since $H$ is a counter-example to the lemma, $H$ has at least $\sum_{v \in V(H)} \omega(v)$ edges. By Lemma~\ref{omega}, $H$ admits a subgraph $S$ that has a good $\omega$-orientation. Let $v$ be a vertex of degree $1$ in $S$, and let $u$ be the neighbour of $v$. We have $\omega(v) \ge 1$, therefore the only edge incident to $v$ goes from $v$ to $u$. Hence for all $w \ne v$, $w$ has the same number of outgoing edges in $S$ and in $S - \{v\}$, so $S - \{v\}$ is a graph with at least one vertex ($u$) and a good $\omega$-orientation. By successively removing vertices of degree $1$ from $S$, we can assume that $S$ is a graph on at least one vertex that admits a good $\omega$-orientation such that there is no vertex of degree $1$ in $S$.
%
%By Lemma~\ref{graphH1}, $S$ has an edge that is not in $L$ and is incident to a big vertex. As no leaf of $L$ is adjacent to a big vertex except for its father, this edge has to link the roots of two connected components of $L$, a contradiction.
%\end{proof}

Let $\widehat L$ be the graph induced by $V(L)$, where we remove every edge that links the roots of two connected components of $L$ and we remove every bud. An \emph{internal} $2$-vertex is a $2$-vertex in $\widehat L$ that has its two neighbours in $\widehat L$. By applying Lemma~\ref{graphH2} to $\widehat L$, we can bound the number of internal $2$-vertices in $\widehat L$. We obtain the following lemma:

\begin{lemm} \label{intern}
The number of internal $2$-vertices is at most $2\sum_{big\ v \in H} (d_G(v)-d-1$).
\end{lemm}

\begin{proof}
Let $\widehat L'$ be the graph $\widehat L$ where every $2$-vertex is removed in the following way: if $v$ is an internal $2$-vertex with neighbours $u$ and $w$, then we remove $v$ and add an edge from $u$ to $w$, and we iterate. For the $2$-vertices that are not internal, we just remove them. Note that $\widehat L'$ may have multiple edges and even loops. As for each $2$-vertex that was removed, exactly one edge was removed, the number of edges in $\widehat L'$ is at most $\sum_{big\ v\in H} \left( d_G(v) -d -1 \right)$. By Lemma~\ref{2v}, every edge of $\widehat L'$ corresponds to at most two internal $2$-vertices. Therefore there are at most $2\sum_{big\ v \in H} (d_G(v)-d-1$) internal $2$-vertices.
\end{proof}

\subsection*{Discharging procedure}
Let $\epsilon = 3 - M$ (recall that $\frac{8}{3} \le M \le 3$). Recall that $d \ge \frac{1}{3-M} = \frac{1}{\epsilon}$, therefore $\epsilon \ge \frac{1}{d} > 0$.

We assign to each $k$-vertex a charge equal to $k - M = k-3 + \epsilon$. Note that since $M$ is bigger than the average degree of $G$, the sum of the charges of the vertices is negative.

Every $3^+$-vertex has a charge of at least $\epsilon > 0$. Therefore every vertex that has a negative charge is a $2$-vertex and has charge $\epsilon - 1$. We will redistribute the weight from the $3^+$-vertices to the $2$-vertices, in order to obtain a non-negative weight on each vertex, by the following three steps:

\begin{enumerate}
\item
Let $S$ be a maximal set of small $3^+$-vertices such that $G[S]$ is connected. Let $S_2$ be a set of $2$-vertices that have exactly one (by Lemma~\ref{2v}) neighbour in $S$. Note that since $\epsilon \le 1$, every $k$-vertex in $S$ has charge at least $(k-2)\epsilon$. The vertices in $S$ give $\epsilon$ to each of the vertices in $S_2$.

Suppose that the total charge of $S$ becomes negative. This implies that the number of vertices in $S_2$ is more than $\sum_{v \in S}(d(v)-2)$. Therefore there are at most $|S|-1$ edges in $G[S]$. Since $G[S]$ is connected, this implies that $G[S]$ is a tree. Now by Lemma~\ref{nobud}, there is at least one big vertex outside of $S$ that has a neighbour in $S$. Note that if there are at least two of these vertices, or if one of them has at least two neighbours in $S$, then one can observe that $G[S]$ has at most $|S|-2$ edges, contradicting the connectivity of $G[S]$. Therefore $S$ is a bud. In this case, $S$ ends up with a charge of at least $-\epsilon$. We will, in Step $2$, make sure that every son of a big vertex in $L$ receives at least $1 - 2\epsilon \ge \epsilon$ (since $\epsilon \le \frac{1}{3}$) from its father, and this will ensure that every bud ends up with a non-negative charge.

We do this step for maximal every set $S$ of small $3^+$-vertices such that $G[S]$ is connected successively. Note that such sets are distinct.

\item
For every big vertex $v$, $v$ gives $1 - 2\epsilon$ to each of its sons, does not give anything to its father (if it has one), and gives $\frac{1 - \epsilon}{2}$ to its other $2$-neighbours. Additionally, every big $k$-vertex gives $2(k - d - 1)\epsilon$ to a common pot.

\item The common pot gives $\epsilon$ to every internal $2$-vertex.
\end{enumerate}

\begin{lemm}\label{nneg2}
Every vertex has non-negative charge at the end of the procedure.
\end{lemm} 

\begin{proof}
Note that by what precedes every small $3^+$-vertex $v$ has non-negative charge.

Every $2$-vertex that is not in $L$ receives $\frac{1 - \epsilon}{2}$ from each of its neighbours. Every light $2$-vertex that is not adjacent to a $2$-vertex (i.e. every $2$-vertex of $L$ that is not an internal $2$-vertex) receives $1-2\epsilon$ from its father and $\epsilon$ from its other neighbour (which is a small $3^+$-vertex). Every internal $2$-vertex  receives $1-2\epsilon$ from its father and $\epsilon$ from the common pot. Therefore every $2$-vertex has charge $0$ at the end of the procedure.

Let us prove that every big vertex has non-negative charge at the end of the procedure. Let $v$ be a big $k$-vertex. Let $c(v)$ be the initial charge of $v$, and $c'(v)$ be the final charge of $v$. Suppose by contradiction that $c'(v)<0$. By Lemma~\ref{kroot}, vertex $v$ has at most $k-1$ sons. Moreover, if $v$ has $k-1$ sons, then its last neighbour is either the father of $v$, or a $3^+$-vertex (by construction of $L$).  Recall that $\epsilon \le \frac{1}{3}$, therefore $1 - 2\epsilon \ge \frac{1-2\epsilon}{2}$.
If $v$ has $k-1$ sons, then $v$ gives $1-2\epsilon$ to each of its $k-1$ sons, and $2(k - d - 1)\epsilon$ to the common pot, therefore $c'(v) = c(v) - (1-2\epsilon)(k-1) + 2(k - d - 1)\epsilon$, and thus as $c'(v) < 0$, we have $c(v) < (1-2\epsilon)(k-2) + 1-\epsilon + 2(k - d - 1)\epsilon$. If $v$ has $k-2$ sons, then it gives $1-2\epsilon$ to each of its $k-2$ sons, and may give at most $\frac{1 - \epsilon}{2}$ to its two other neighbours and $2(k - d - 1)\epsilon$ to the common pot, therefore $c(v) < (1-2\epsilon)(k-2) + 1-\epsilon + 2(k - d - 1)\epsilon$. If we decrease the number of sons of $v$ further than $k-2$, we will still have $c(v) < (1-2\epsilon)(k-2) + 1-\epsilon + 2(k - d - 1)\epsilon$.

Thus if $c(v) \ge (1-2\epsilon)(k-2) + 1-\epsilon + 2(k - d - 1)\epsilon$, we get a contradiction. Recall that $c(v)$ is equal to $k-3 + \epsilon$.
Therefore we only need to prove that $k-3 + \epsilon\ge (1-2\epsilon)(k-2) + 1-\epsilon + 2(k - d - 1)\epsilon$, which is equivalent to $d \ge \frac{1}{\epsilon}$.
\end{proof}

Let us prove that the common pot also has non-negative charge at the end of the procedure. It receives charge $\sum_{v \text{ big}}2(d(v) - d - 1)\epsilon$. By Lemma~\ref{intern}, this charge is at least $\epsilon$ times the number of internal $2$-vertices. The common pot gives $\epsilon$ to each internal $2$-vertex, therefore it has non-negative charge at the end of the procedure.


By Lemma~\ref{nneg2}, every vertex has non-negative charge at the end of the procedure, thus the sum of the charges at the end of the procedure is non-negative. Since no charge was created nor removed, and since the common pot also has non-negative charge, this is a contradiction with the fact that the initial sum of the charges is negative. That ends the proof of Theorem~\ref{t-main2}.

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