Weak degeneracy of planar graphs and locally planar graphs

Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d + 1)$-list assignment $L$ of $G$, one can construct an $L$-coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are 5-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate.


Introduction
For a graph G, the greedy coloring algorithm colors vertices one by one in order v 1 , v 2 , . . ., v n , assigning v i the least-indexed color not used on its colored neighbors.An upper bound for the number of colors used in such a coloring is captured in the notion of graph degeneracy.Let Z be the set of integers, and Z G be the set of mappings f : V (G) → Z.For f ∈ Z G and a subset U of V (G), let f | U be the restriction of f to U , and let f −U : V (G) − U → Z be defined as f −U (x) = f (x) − |N G (x) ∩ U | for x ∈ V (G) − U .For convenience, we may use f for f | U , and write f −v for f −{v} .We denote by E[U ] the set of edges in G with both end vertices in U .
Let L be the set of pairs (G, f ), where G is a graph and f ∈ Z G .
Definition 1.The deletion operation Delete(u) : L → L is defined as We say Delete(u) is legal for (G, f ) if both f and f −u are non-negative.A graph G is f -degenerate if, starting with (G, f ), it is possible to remove all vertices from G by a sequence of legal deletion operations.For a positive integer d, we say that G is d-degenerate if it is degenerate with respect to the constant d function.The degeneracy of G, denoted by d(G), is the minimum d such that G is d-degenerate.
The quantity d(G) + 1 is called the coloring number of G, and is an upper bound for many graph coloring parameters: the chromatic number χ(G), the choice number χ (G), the paint number χ P (G), the DP-chromatic number χ DP (G) and the DP-paint number χ DPP (G).The definitions of some of these parameters are complicated.As we shall not discuss these parameters, other than saying that they are bounded by the weak degeneracy defined below, we omit the definitions and refer the reader to [6] for the definitions and discussion about these parameters.
The coloring number d(G) + 1 of G, as an upper bound for the above mentioned graph coloring parameters, is often not tight.It is therefore interesting to see if we can modify the greedy coloring algorithm to save some of the colors and get a better upper bound.Motivated by this, Bernshteyn and Lee [1] recently introduced the concept of weak degeneracy of a graph.
Assume L is a list assignment of G and we try to construct an L-coloring of G. Assume uw is an edge of G.In the greedy coloring algorithm, if we assign a color to u, then it is counted that L(w) loses one color.However, if |L(u)| > |L(w)|, then one can assign to u a color from L(u) − L(w), and hence L(w) will not lose a color in this step.The concept of weak degeneracy deals with this situation.
Definition 2. The deletion-save operation DeleteSave(u, w) : L → L is defined as and both f and f −u + δ w are non-negative.Definition 3. A removal scheme Ω = Del(θ 1 , θ 2 , . . ., θ k ) : L → L, where for each i, either θ i = u i representing the deletion operation Delete(u i ), or θ i = u i , w i representing the deletionsave operation DeleteSave(u i , w i ), is defined recursively as follows: ) and removes all vertices of G.For a positive integer d, we say that G is weakly d-degenerate if it is weakly degenerate with respect to the constant d function.The weak degeneracy of G, denoted by wd(G), is the minimum d such that G is weakly d-degenerate.
The following proposition was proved in [1].
Some well-known upper bounds for χ DP (G) for families of graphs turn out to be upper bounds for wd(G) + 1.For example, Bernshteyn and Lee [1] proved that planar graphs are weakly 4degenerate and Brooks theorem remains true for weak degeneracy.
It was proved by Thomassen [8] that planar graphs of girth at least 5 are 3-choosable.Dvořák and Postle [4] observed that planar graphs with girth at least 5 are DP-3-colorable.This paper strengthens this result and show that planar graphs of girth at least 5 are weakly 2-degenerate.Indeed, we shall prove graphs in a slightly larger graph family are weakly 2-degenerate.
We write P = v 1 v 2 . . .v s to indicate that P is a path with vertices v 1 , v 2 , . . ., v s in this order, and write K = (v 1 v 2 . . .v k ) to indicate that K is a cycle with vertices v 1 , v 2 , . . ., v k in this cyclic order.For convenience, we also denote by P and K the vertex sets of P and K, respectively.The length of a path or a cycle is the number of edges in the path or cycle.A k-cycle (respectively, a k − -cycle or a k + -cycle) is a cycle of length k (respectively, at most k or at least k).Two cycles are adjacent if they share some common edges, and we say they are normally adjacent if their intersection is isomorphic to K 2 .Let G denote the class of triangle-free plane graphs in which no 4-cycle is normally adjacent to a 5 − -cycle.Dvořák, Lidický and Škrekovski [3] proved that every graph in G is 3-choosable.In this paper, we prove that every graph in G is weakly 2-degenerate.The proof uses induction, and for this purpose, we prove a stronger and more technical result.
For a plane graph and a cycle K, we use int(K) to denote the set of vertices in the interior of K, and ext(K) to denote the set of vertices in the exterior of K. Denote by int[K] and ext[K] the subgraph of G induced by int(K) ∪ K and ext(K) ∪ K, respectively.For the plane graph G, we denote by B(G) the boundary walk of the infinite face of G.
Theorem 1.Let G ∈ G, and P = p 1 p 2 . . .p s be a path on B(G) with at most four vertices.Let f ∈ Z G be a function satisfying the following conditions: ∈ B(G), and 1 ≤ f (v) ≤ 2 for all v ∈ B(G)\V (P ); is an independent set in G, and each vertex in I has at most one neighbor in P .
The following is an easy consequence of Theorem 1.
Corollary 1.Every graph in G is weakly 2-degenerate.In particular, every planar graph of girth at least 5 is weakly 2-degenerate.
The proof of Theorem 1 uses induction, and follows a similar line as the proof of the 3choosability of these graphs in [3].Indeed, the idea of DeleteSave operation was used in some cases in [3] (as well as in many other papers on list colouring of graphs), although the term DeleteSave was not used explicitly.Nevertheless, the proof of Theorem 1 requires rather different treatments in some cases.The conclusion that these graphs are weakly 2-degenerate is intrinsically stronger.For example, it implies that these graphs are DP 3-paintable, and the proof in [3] does not apply to DP-coloring.
Assume S is a surface and G is a graph embedded in S. A cycle C in G is contractible if, as a closed curve on S, it separates S into two parts, and one part is homeomorphic to the disc.We say C is non-contractible otherwise.The length of the shortest non-contractible cycle in G is called the edge-width of G and is denoted by ew(G).Note that if S is the sphere, then every closed curve in S is contractible, and hence ew(G) = ∞ for any graph G embedded in S. We say a graph G embedded in a surface S is "locally planar" if ew(G) is "large".It was proved by Thomassen [7] that for any surface S, there is a constant w such that any graph G embedded in S with ew(G) ≥ w is 5-colorable.Roughly speaking, this result says that locally planar graphs are 5-colorable.This result was strengthened in a sequence of papers, where it was proved that locally planar graphs are 5-choosable [2], 5-paintable [5] and DP 5-paintable [6].In this paper, we further strengthen this result by proving the following result.
Theorem 2. For any surface S, there is a constant w(S) such that every graph G embedded in S with edge-width at least w(S) is weakly 4-degenerate.

Some preliminaries
For If Ω = Del(θ 1 , . . ., θ k ) and for each i, either θ i = u i or θ i = u i , w i , then let U Ω = {u 1 , u 2 , . . ., u k }.Note that for any removal scheme Ω, we have Observation 2. The following follows from the definition.
and removes all the vertices of G − v, then Ω = Del( v , θ 1 , . . ., θ n ) is legal for (G, f ) and removes all the vertices of G since Conversely, assume that Ω = Del(θ 1 , . . ., θ n ) is legal for (G, f ) that removes all the vertices of G.As f (v) = 0, v is removed by a deletion operation and so there is an index i such that θ i = v .For j < i, let Note that if uv ∈ E(G), and u is removed in a move θ j for some j < i, then since f (v) = 0, we must have θ j = u, v .By repeatedly applying Observation 2, we conclude that and removes all vertices of G − v.

Proof of Theorem 1
It follows from Proposition 2 that the conclusion of Theorem 1 is equivalent to G − P is weakly f −P -degenerate.In the proof below, for different cases, we shall prove either of these two statements.
Definition 4. Assume G is a plane graph and P is a boundary path, and f ∈ Z G .We say It follows from the definition and Proposition 2 that if Ω is legal for (G, P, f ), and Assume Theorem 1 is not true, and (G, P, f ) is a counterexample with minimum |V (G)| + |E(G)|, and subject to this, with minimum v∈V (G)\V (P ) f (v).To derive a contradiction, it suffices to find a removal scheme Ω legal for (G, P, f ), so that (G Ω , P, f Ω ) satisfies the condition of Theorem 1.Note that Ω is required to be legal for (G, P, f ), i.e., legal for (G − P, f −P ), and is not required to be legal for (G, f ).On the other hand, Ω is applied to (G, f ), and G Ω contains the path P .Alternately, we may apply Ω to (G − P, f −P ).Then to apply the induction hypothesis to the resulting graph, we need to change (G − P ) Ω back to G Ω (i.e., add back the path P ) and change (f Assume P G 1 and P G 2 .Let P 1 = P ∩ G 1 and P 2 = P ∩ G 2 .Then v ∈ V (P ), and For convenience, assume k = 7 (the k ≤ 6 can be treated similarly), and assume that It is easy to verify that (G , P, f ) satisfies the conditions of Theorem 1, and hence G − P is weakly f −P -degenerate.As , and B(G) − P is certainly weakly f −P -degenerate, it follows from Observation 1 that G − P is weakly f −P -degenerate.
Next we assume that K is a separating By the minimality of G, G − P is weakly f −P -degenerate.The same argument as above shows that int[K] − K is weakly f −K -degenerate.A k-chord of B(G) is a path Q of length k such that only its two ends are on B(G).A 1-chord is also called a chord of B(G).Let G 1 , G 2 be the two subgraphs with . We say G 1 and G 2 are the subgraphs of G separated by Q.We index the subgraphs so that It is obvious that (G 1 , P, f ) satisfies the conditions of Theorem 1, and hence G 1 − P is weakly f −P -degenerate.If P 2 is an induced path and (G 2 , P 2 , f ) also satisfies the conditions of Theorem 1, then G 2 − P 2 is weakly f −P 2 -degenerate, and it follows from Observation 1 that G − P is weakly f −P -degenerate, a contradiction.Thus we may assume that (G 2 , P 2 , f ) does not satisfy the conditions of Theorem 1.

Lemma 4. B(G) has no chords.
Proof.Assume to the contrary that B(G) has a chord uw.Let G 1 , G 2 be the two subgraphs of G separated by uw.
Assume P ⊆ G 1 .Since G is triangle-free, each vertex in G 2 is adjacent to at most one vertex in {u, w}.Thus (G 2 , uw, f ) satisfies the conditions of Theorem 1, in contrary to Observation 3.
Assume P G 1 and P G 2 .Without loss of generality, assume that w ∈ V (P ).Then ∈ V (P ) and P is an induced path.We may assume that |E(P ∩ G 2 )| = 1.If P 2 is not contained in a 4-cycle in G 2 , then (G 2 , P 2 , f ) satisfies the conditions of Theorem 1, a contradiction.
Assume P 2 is contained in a 4-cycle in G 2 .Since no 4-cycle in G is adjacent to a 5 − -cycle, uw is not contained in a 5 − -cycle in G 1 .Let P 1 = uw ∪ (P ∩ G 1 ).It is easy to verify that (G 2 , P, f ) and (G 1 , P 1 , f ) satisfy the conditions of Theorem 1.By the minimality of G, G 2 − P is weakly f −P -degenerate, and G 1 − P 1 is weakly f −P 1 -degenerate.It follows from Observation 1 that G − P is weakly f −P -degenerate, a contradiction.Since B(G) is an induced cycle of length at least 8 and (G, P, f ) is a counterexample with minimum v∈V (G)\V (P ) f (v), we may assume that P = p 1 p 2 p 3 p 4 is an induced path of length three.Assume B(G) = p 1 p 2 p 3 p 4 x 1 x 2 . . .x m , where m ≥ 4. We say a k-chord Q of B(G) splits off a face F from G if one of the two subgraphs separated by Q is the boundary cycle of F .Lemma 5. Let uvw be a 2-chord of B(G).Then {u, w} V (P ), and uvw splits off a 5 − -face Consequently, every internal vertex is adjacent to at most two vertices in B(G) and adjacent to at most one vertex in V (P ).
Proof.Assume uvw is a 2-chord and G 1 and G 2 are subgraphs separated by uvw.Assume and hence P 2 has length 2 or 3.If P 2 is not induced, then since G is triangle-free and B(G) has no chord, G[P 2 ] is a 4-cycle that bounds a 4-face by Lemma 2. So we may assume P 2 is an induced path.
By Observation 3, (G 2 , P 2 , f ) does not satisfy the conditions of Theorem 1.This means that G 2 has a vertex y with f (y) = 1 and y is adjacent to two vertices of P 2 .
If P 2 has length 2, then G 2 is a 4-cycle, and hence uvw splits off a 4-face.Moreover, {u, w} ⊆ V (P ), for otherwise, G 1 is a 5 − -cycle, in contrary to Lemma 3.
Assume P 2 has length 3. We may assume w = p 3 and P 2 = uvp 3 p 4 .As B(G) has no chord, we know that either y is adjacent to p 4 and v, or y is adjacent to p 4 and u.
If y is adjacent to p 4 and v, then u / ∈ P (for otherwise G 1 is a 4-cycle and G contains two adjacent 4-cycles).Then yvu is a 2-chord which separates G into G 1 and G 2 with P ⊆ V (G 1 ).Let P 2 = yvu.Then (G 1 , P, f ) and (G 2 , P 2 , f ) satisfy the conditions of Theorem 1, and hence G 1 − P is weakly f −P -degenerate, and G 2 − P 2 is weakly f −P 2 -degenerate (note that in this case, G 2 is not a 4-cycle as G contains no two adjacent 4-cycles).By Observation 1, G − P is weakly Assume y is adjacent to p 4 and u.Then G 2 is a facial 5-cycle by Lemma 2. If u ∈ P , then u = p 1 and G 1 is a facial 4-cycle by Lemma 2, implying that deg G (v) = 2, a contradiction.Thus u ∈ P and uvw splits off a 5-face.
Proof.Let G 1 and G 2 be the two subgraphs of G separated by Q.Since {u, z} ∩ {p 2 , p 3 } = ∅, we may assume that P ⊂ G 1 .
If uz ∈ E(G), then since B(G) has no chord, G 2 is a facial 4-cycle and Q splits off a 4-face.Assume uz / ∈ E(G).By Observation 3, (G 2 , Q, f ) does not satisfy the conditions of Theorem 1. Then there exists a vertex x with f (x) = 1 adjacent to two vertices in {u, v, w, z}.If x is adjacent to u and z, the Q splits off the 5-face F bounded by uvwzxu.If x is adjacent to u and w, then by Lemma 5, the 2-chord xwz splits off a 4-face [xwzz x].Then uvwxu and xwzz x are adjacent 4-cycles, which contradicts Lemma 3. The case that x is adjacent to z and v is symmetric.
We may assume that f (x 1 ) = 1 or f (x 2 ) = 1, for otherwise, let f = f except that f (x 1 ) = 1, (G, P, f ) satisfies the conditions of Theorem 1.Note that I is independent in G. Hence by the minimality of v∈V (G)\V (P ) f (v), G − P is weakly f −P -degenerate, and hence G − P is weakly f −P -degenerate.
We write f (x 1 , x 2 , . . ., x i ) = (a 1 , a 2 , . . ., a i ) to mean that f (x j ) = a j for j = 1, 2, . . ., i. Let X be the set of boundary vertices defined as follows: boundary of D .Then each vertex v ∈ B(D ) is adjacent to at most two neighbors in G 0 , and so f −G 0 (v) ≥ 2. Every interior vertex v ∈ V (D − B(D )) is not adjacent to any vertex of G 0 , and hence f −G 0 (v) = 4.By Lemma 9, D is weakly f −G 0 -degenerate.By Observation 1, G 0 is weakly f -degenerate, a contradiction.
It remains to prove Theorem 3. We may assume that G is a triangulation of the surface S, because adding edges does not decrease the face-width or the connectivity of a graph.First, we will recall the definition of nice H-scheme in [5], which is an important structure in our proof.If the vertices u e,x and/or u e,y do not exist, then ignore them in the above formula.Let P (e, x) = P (e, x) ∪ {v − e,x , u − e,x }.A segment of a path P is subset of its vertices that induces a subpath of P .
Assume F is an H-scheme in a graph G embedded in S. ( (3) For e ∈ E(H), x ∈ e and v ∈ U , if N R (v) ∩ P (e, x) = ∅, then N R (v) ∩ V (G) is a segment of P (e, x), and if av is an in-edge of v, then v has two out-edges vb, vc such that b, c lies between v e,x and a on P (e, x).
Proof of Theorem 3. Assume S is a surface, w(S) is the constant in Lemma 10, G is a 5connected triangulation of S with face-width at least w(S).Let F be a H-scheme in G, and R be the associated bipartite graph oriented as in Lemma 10.Let f (u) = 4 for any vertex u ∈ V (G), and let Note that each component of G 2 is a path (P (e, x)∪{u − e,x })\{v e,x } for some e ∈ E(H), x ∈ e.We order the vertices of G 2 as x 1 , x 2 , . . ., x t so that if i < j and x i , x j ∈ (P (e, x)∪{u − e,x })\{v e,x }, then x i lies between v e,x and x j on P (e, x).By (A2) of Definition 5, f −G 1 (x i ) = 3 if x i is the unique vertex of P (e, x) adjacent to v e,x and f −G 1 (x i ) = 4 otherwise.By Lemma 10, . We define a removal scheme Ω = Del(θ 1 , θ 2 , . . ., θ t ) as follows: for i ∈ [t], Let h be the function produced by legal removing all the vertices in 4 i=1 G i .Finally, for each vertex u on the boundary of G 5 , h(u) ≥ 2 since the value of f (u) only decreases when its out-neighbor is removed and u has at most two out-neighbors in 4 i=1 G i .For the interior vertex u, h(u) = f (u) = 4. Similarly, by Lemma 9, G 5 is weakly h-degenerate.
As a consequence of Theorem 2, the condition in Theorem 3 on the connectivity is redundant.
Corollary 2. For any surface S there is a constant w(S) such that every graph G embedded in S with face-width at least w(S) is weakly 4-degenerate.

Lemma 2 .
|B(G)| ≥ 8 and every separating cycle in G has length at least 8. Proof.Assume B(G) = (v 1 v 2 . . .v k ) for some k ≤ 7.As G is triangle-free and no 4-cycle is normally adjacent to a 5 − -cycle, B(G) is an induced cycle.

Lemma 3 .
There are no 4-cycles adjacent to 4-or 5-cycles.Proof.Suppose to the contrary that a 4-cycle C 1 is adjacent to a 5 − -cycle C 2 .By assumption, C 1 and C 2 are not normally adjacent.So they intersect at three vertices.As C 2 has no chord, we may assume that C 1 = [a 1 a 2 a 3 a 4 ] and C 2 = [a 1 a 2 a 3 b 4 ] or C 2 = [a 1 a 2 a 3 b 4 b 5 ].By Lemma 2, each of C 1 and C 2 bounds a face.Thus a 2 is a 2-vertex which must be on the outer face.But then [a 1 a 4 a 3 b 4 ] or [a 1 a 4 a 3 b 4 b 5 ] is a separating cycle of length at most 5, contradicting Lemma 2.

Definition 5 .
Assume G is a graph embedded in S and H is a cubic graph.An H-scheme in G is a family F of induced subgraphs of G together with a labeling which associates subgraphs in F to vertices and edges of H such that the following hold:A1 F = {D(x) : x ∈ V (H)} ∪ {D(e) : e ∈ E(H)} ∪ {P (e,x) : e ∈ E(H), x ∈ e} consists of a family of subgraphs of G, each embedded in a disk in S, and for e ∈ E(H) and x ∈ e, P (e, x) is a path connecting a vertex v e,x on the boundary of D(x) to a vertex u e,x on the boundary of D(e).A2 The subgraphs in F are pairwise disjoint, except that v e,x belong to both P (e, x) and D(x), and u e,x belong to both P (e, x) and D(e).Also no edge of G connects vertices of distinct subgraphs in F, except that v e,x has neighbors in both D(x) and P (e, x), and u e,x has neighbors in both D(e) and P (e, x).A3 By contracting each D(x) into a single vertex for each x ∈ V (H), and replace each P (e, x)∪ D(e) ∪ P (e, y) by an edge joining x and y for each edge xy ∈ E(H), we obtain a 2-cell embedding of H in S. For e = xy ∈ E(H), let v − e,x and v + e,x be the two neighbors of v e,x on the boundary of the outer face of D(x), u − e,x and u + e,x be the two neighbors of u e,x on the boundary of the outer face of D(e), and u e,x be the unique common neighbor of u e,x , u + e,x , u − e,x in D(e), if such a vertex exists.Let D (e) = D(e) − {u e,x , u − e,x , u + e,x , u e,x , u e,y , u − e,y , u + e,y , u e,y }.
each component of G is a plane graph embedded in a disk on S. Let R be the bipartite subgraph of G induced by edges between U and U .The following lemma is a combination of Lemma 2.4 and Lemma 3.2 in[5].Lemma 10.For any surface S, there is a constant w(S) such that the following holds: If a 5-connected triangulation G of S has face-width at least w(S), then there is a cubic graph H such that G has an H-scheme F satisfying for each edge e = xy of H, for any vertex u ∈ D (e), N G (u) ∩ U ⊆ D(e), |N G (u) ∩ {u e,x , u + e,x , u − e,x , u e,x , u e,y , u + e,y , u − e,y , u e,y }| ≤ 2, dist G (u e,x , u e,y ) ≥ 5.Moreover, the associated bipartite graph R has an orientation for which the following hold: (1) For v ∈ U , deg + R (v) ≤ 1.Moreover, for e ∈ E(H) and x ∈ e, if v ∈ D(x) or v ∈ D(e) − {u e,x , u − e,x , u e,y , u − e,y }, then deg + R (v) = 0;

3 i=1V
and so it is legal.Otherwise, deg + R (x i ) = 1.By (3) of Lemma 10, w i has two removed out-neighbors.Thus,f −(G 1 ∪A i−1 ) (w i ) ≤ 2 < 3 ≤ f −(G 1 ∪A i−1 ) (x i ),and so it is also legal.Now we consider the vertices in G 3 .By definition and Lemma 10, each component of G 3 has at most two vertices, u + e,x and u e,x (if exists).As for each e ∈ E(H), x ∈ e, u + e,x has only one neighbor in G 1 ∪ G 2 and u e,x (if exists) has two removed neighbors inG 1 ∪ G 2 , f −(G 1 ∪G 2 ) (v) ≥ 2 for v ∈ G 3 .Then for each component, we can remove its vertices by deletion operation in the order u + e,x , u e,x .With the same operation, we can remove all the vertices of G 3 legally since the components of G 3 do not affect each other.In the subgraph G 4 , as each component of G 4 is a plane graph D (e) for some e ∈ E(H), G 4 is a plane graph.By Lemma 10, for each edge e = xy of H, for any vertex u ∈ D (e),N G (u) ∩ (G i ) ⊆ {u e,x ,u + e,x , u − e,x , u e,x , u e,y , u + e,y , u − e,y , u e,y }, and |N G (u) ∩ {u e,x , u + e,x , u − e,x , u e,x , u e,y , u + e,y , u − e,y , u e,y }| ≤ 2. Therefore, f −( 3 i=1 G i ) (u) ≥ 2 if u is on the boundary of G 4 , and f −( 3 i=1 G i ) (u) = 4 otherwise.By Lemma 9, G 4 is weakly f −( 3 i=1 G i )degenerate.