Weakening Total Coloring Conjecture: Weak TCC and Hadwiger's Conjecture on Total Graphs

Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\sqcup E(G)$ with $c_1,c_2\in V(G)\sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwiger's conjecture is true for $T(G)$. The total chromatic number $\chi"(G)$ of a graph $G$ is defined to be equal to the chromatic number of its total graph. That is, $\chi"(G)=\chi(T(G))$. Another well-known conjecture in graph theory, the total coloring conjecture or TCC, states that for every graph $G$, $\chi"(G)\leq\Delta(G)+2$, where $\Delta(G)$ is the maximum degree of $G$. We show that if a weaker version of the total coloring conjecture (weak TCC) namely, $\chi"(G)\leq\Delta(G)+3$, is true for a class of graphs $\mathcal{F}$ that is closed under the operation of taking subgraphs, then Hadwiger's conjecture is true for the class of total graphs of graphs in $\mathcal{F}$. This motivated us to look for classes of graphs that satisfy weak TCC. It may be noted that a complete proof of TCC for even 4-colorable graphs (in fact even for planar graphs) has remained elusive even after decades of effort; but weak TCC can be proved easily for 4-colorable graphs. We noticed that in spite of the interest in studying $\chi"(G)$ in terms of $\chi(G)$ right from the initial days, weak TCC is not proven to be true for $k$-colorable graphs even for $k=5$. In the second half of the paper, we make a contribution to the literature on total coloring by proving that $\chi"(G)\leq\Delta(G)+3$ for every 5-colorable graph $G$.


Introduction 1.Total Coloring Conjecture
Let G be a simple finite graph.For a vertex v ∈ V (G) in G, we define N G (v) := {u ∈ V (G) : u is adjacent to v in G} and E G (v) := {e ∈ E(G) : e is incident on v in G }.The degree of a vertex v ∈ V (G) is defined as A vertex coloring is called a proper vertex-coloring if no two adjacent vertices are assigned the same color.The minimum number of colors required to achieve a proper vertex-coloring of G is called the chromatic number of G, and is denoted by χ(G).Like the vertex-coloring problem, the problem of coloring the edges of a graph is also one that has received much attention in the literature.A coloring of the edges of a graph is called a proper edge-coloring if no two adjacent edges are assigned the same color.The minimum number of colors required in any proper edge-coloring of a graph G is called its edge-chromatic number or chromatic index, and is denoted by χ ′ (G).The following theorem was proved by Vizing : Theorem 1.1 (Vizing's Theorem).For a simple finite graph G, ∆(G) ≤ χ ′ (G) ≤ ∆(G) + 1.
Vizing's theorem suggests a classification of simple finite graphs into two classes.A graph G is said to be in class I if χ ′ (G) = ∆(G) and in class II if χ ′ (G) = ∆(G) + 1.
The notion of line graphs allows us to view the edge coloring problem as a vertex coloring problem.Given a graph G, the line graph of G is the graph L(G) = (V ′ , E ′ ) where V ′ = E(G) and e 1 , e 2 ∈ V ′ form an edge e 1 e 2 ∈ E ′ whenever e 1 and e 2 are adjacent edges in G. Clearly χ ′ (G) = χ(L(G)).The clique number of G, denoted by ω(G), is the number of vertices in a maximum clique of G. Clearly ω(G) ≤ χ(G) for all graphs.It is easy to see that ω(L(G)) ≥ ∆(G), and therefore Vizing's theorem guarantees that the chromatic number of any line graph is at most 1 more than its clique number.Given a class of graphs F , if there exists a function f such that ∀G ∈ F , χ(G) ≤ f (ω(G)), then F is a χ-bounded class and f is a χ-bounding function for F .χ-boundedness of graph classes is an extensively studied topic, see the survey by Scott and Seymour [26] for further references.Thus Vizing's Theorem guarantees a very strong χ-bounding function for the class of line graphs.
After vertex and edge colorings it was natural for graph theorists to consider coloring vertices and edges simultaneously.A total coloring of a graph G is a coloring of all its elements, that is, vertices and edges, such that no two adjacent elements are assigned the same color.(When we refer to two adjacent elements of a graph, they are either adjacent to each other or incident on each other as the case may be.)That is, it is an assignment c of colors to V (G) ∪ E(G) such that c| V (G) is a proper vertex-coloring, c| E(G) is a proper edge-coloring, and c(uv) / ∈ {c(u), c(v)} for any edge uv ∈ E(G).The minimum possible number of colors in any total coloring of G is called the total chromatic number of G, and is denoted by χ ′′ (G).
The total coloring conjecture (or TCC ) was proposed by Behzad [3] and Vizing [27] independently between 1964 and 1968.Remark.If we use multigraphs instead of simple graphs, the above statement may not hold.This is true of Theorem 1.1 also.In this paper we consider only simple finite graphs.
Even though many researchers have examined TCC over the years, it remains unsolved till date and is considered one of the hardest open problems in graph coloring.Bollobás and Harris [6] proved that χ ′′ (G) ≤ 11  6 ∆(G), when ∆(G) is sufficiently large.Later Kostochka [17] proved that χ ′′ (G) ≤ 3 2 ∆(G), when ∆(G) ≥ 6.The best known result to date for the general case was obtained by Molloy and Reed [19] : Remark.Molloy and Reed have mentioned in [19] that though they only show a value 10 26 for the constant C, with much more effort it can be brought down to 500.
While TCC remains unsolved for the general case, it is known to hold for some special classes of graphs.For example, it is easy to see that TCC is true for all complete graphs and bipartite graphs [4].Another case is the class of graphs with maximum degree at most 5.
For planar graphs, the total coloring conjecture is known to hold except when the maximum degree is 6.
Considering the difficulty level of TCC, it makes sense to study relaxations of TCC.As the title of the paper indicates, we intend to consider two ways to weaken TCC.First one is the most obvious way: Increase the upper bound appearing in the statement of TCC.This leads to weak-TCC and (k)-TCC; see Section 2 for more details.Now we will develop the background for the second one.
Just like we defined line graphs and used it to perceive the edge coloring problem on G as a vertex coloring problem on L(G), we can define a similar structure in the context of total coloring also.Given a graph G = (V, E) the total graph of G is the graph T (G) = (V ′′ , E ′′ ) where V ′′ = V ∪ E, the set of all elements of G, and c 1 c 2 ∈ E ′′ whenever elements c 1 ,c 2 are adjacent in G. Given a class F of graphs, we define T (F ) := {T (G) : G ∈ F }. We use T to denote the class of total graphs i.e.T = T (G), where G is the class of all simple finite graphs.
Note.We shall call vertices in T (G) that correspond to vertices in the original graph G as v -vertices and vertices that correspond to edges in G as e-vertices.For ease of notation, we denote an element of G, that is, an edge or a vertex in G, and its corresponding vertex in T (G), by the same letter.Thus, the v -vertex in T (G) corresponding to x ∈ V (G) is also denoted by x.
Observation 1.Any total coloring of a graph G corresponds to a proper vertex coloring of T (G) and viceversa.Thus, the total chromatic number of G is equal to the chromatic number of T (G) that is, χ ′′ (G) = χ(T (G)).
Observation 2. For every vertex x ∈ V (G), the v -vertex corresponding to x and the e-vertices corresponding to the edges in E G (x) form a clique of order d G (x) + 1 in T (G).It follows that there exists a clique of order ∆(G) + 1 in T (G).Therefore, ∆(G) + 1 ≤ ω(T (G)) ≤ χ(T (G)) = χ ′′ (G).In fact, it is not difficult to see that the clique number of the total graph of G, ω(T (G)) = ∆(G) + 1, when ∆(G) ≥ 2.
It is easy to see that if ∆(G) ≤ 1, then χ ′′ (G) = ω(T (G)).Thus, in view of Observation 2, Conjecture 1.2 can be thought to be suggesting the existence of a very strong χ-bounding function for the class of total graphs, and can be restated as follows: The total coloring conjecture was intensely studied by several brilliant graph theorists for more than 50 years, but still remains unsolved and has earned the reputation of being one of the toughest problems in graph coloring.Therefore it makes sense to study relaxed versions of this question.From the structural perspective, one obvious way to relax Conjecture 1.6 is to replace clique number ω(H) by Hadwiger number η(H), the number of vertices in the largest clique minor of H.This directly leads us to the study of Hadwiger's conjecture on Total Graphs, hereafter abbreviated as "HC on T ". (See next section for the formal statement of Hadwiger's Conjecture.)Problem 1.7 (Hadwiger's Conjecture on Total Graphs or "HC on T ").

χ(H) ≤ η(H), ∀H ∈ T
Remark.Careful readers may object that if we replace ω(H) by η(H) in Conjecture 1.6, the new inequality should be χ(H) ≤ η(H) + 1, ∀H ∈ T .But we will show in Theorem 2.4 that Conjecture 1.6 indeed implies HC on T as stated in Problem 1.7.
Needless to say, Hadwiger's conjecture is an even more celebrated and long standing conjecture than TCC in graph theory, owing its origin to the Four Color Theorem itself.While the general case of Hadwiger's conjecture remains unsolved, it was proved for several special classes of graphs.For example, Reed and Seymour [20] proved Hadwiger's conjecture for line graphs more than 15 years ago.Chudnovsky and Fradkin [10] even generalized the result to quasi-line graphs.But to the best of our knowledge, for the class of total graphs, whose definition is similar in spirit to that of line of graphs, Hadwiger's conjecture has not yet been proved or studied.
In other words, every graph has either a clique minor on t vertices (K t -minor) or a proper vertex-coloring using (t − 1) colors.
For t ≤ 3, Hadwiger's conjecture is easy to prove.The t = 4 case was proved by Hadwiger himself in [13].The Four Color Theorem was proved by Appel and Haken [1,2] in 1977.Using a result of Wagner [29], it can be shown that the Four Color Theorem is equivalent to Hadwiger's conjecture for t = 5.Robertson, Seymour, and Thomas [21] proved in 1993 that Hadwiger's conjecture holds true for t = 6.The conjecture remains unsolved for t ≥ 7.So far, Hadwiger's conjecture has been proved for several classes of graphs; see [5], [10], [18], [20], [30], [31].Now we describe a curious observation from [8].Given a graph G, its square G 2 is defined on the vertex set V (G) with u and v being adjacent in G 2 whenever the distance between u and v in G is at most 2. For a class of graphs F , let F 2 denote the set of graphs {G 2 : G ∈ F }. Recall that a split graph is a graph whose vertex set can be partitioned into an independent set and a clique.Let S denote the special class of split graphs, whose vertex set is partitioned into an independent set A and a clique B with the following extra constraints: (1) Each vertex in B has exactly 2 neighbours in A. (2) There are no two vertices in B having the same neighbourhood in A.
The following fact is from [8].(In [8], the following fact is not stated explicitly in this detail, but can be easily read out from the proof of Theorem 1.2, therein.)Fact.Proving the general case of Hadwiger's conjecture is equivalent to proving HC for the class S 2 .
Obviously proving HC for S 2 is extremely difficult, despite its tantalizingly specialized appearance.From this perspective, it is natural to consider classes of graphs that can be obtained by simple structural modifications of S and see how difficult it is to prove HC for the squares of such classes.One such modification is to assume that both A and B are independent sets, keeping everything else same, i.e. the class of bipartite graphs with parts A and B with the extra constraints: (1) Each vertex in B has exactly 2 neighbours in A. (2) No two vertices in B have the same neighbourhood in A. In other words get a bipartite graph from each split graph in S by converting the clique B to an independent set.Let this new class be denoted by Ŝ.How difficult is it to prove HC for Ŝ2 ?Careful inspection reveals that Ŝ is a familiar class of graphs, the sub-divided graphs.
A graph G is a subdivided graph if it can be obtained from another graph H by subdividing each edge uv in H; that is, replacing uv by a path uwv, where w is a new vertex.Now if we consider the set of newly introduced vertices as A and the original vertices of H as B, it is easy to see that G ∈ Ŝ.The converse is also true: it is easy to verify that if a graph G belongs to Ŝ, then there exists another graph H such that G is obtained by sub-diving all the edges of H. Interestingly, the class of squares of subdivided graphs is the same as the class of total graphs.
Fact.If H is a graph and G is obtained by subdividing every edge of H, then the graph G 2 is isomorphic to the total graph T (H) of H.In other words, the class of total graphs T = Ŝ2 .
Considering the structural closeness of Ŝ and S and the fact from [8] that HC on S 2 is equivalent to the general case of HC, it is not very surprising that proving Hadwiger's Conjecture for total graphs turns out be challenging.
Remark.Let G be a graph and let T (G) be its total graph.Denote the set of v -vertices and the set of e-vertices in T (G) by V and E, respectively.Note that the subgraph of T (G) induced by V is isomorphic to the original graph G, the subgraph of T (G) induced by E is isomorphic to the line graph L(G) of G, and the bipartite graph induced by the edges between the sets V and E is precisely the subdivided graph S(G) of G.

Our contributions to Hadwiger's conjecture on total graphs
In Section 2 and Section 3, we explore the difficulty level of HC on T in comparison with TCC.Our first result is Theorem 2.4 which states that if TCC is true for a class F of graphs that is closed under taking subgraphs, then Hadwiger's conjecture holds true for the class T (F ).Taking F = G, the class of simple finite graphs, we infer that TCC implies HC on T , thus justifying our claim that Problem 1.7 is a relaxation of Conjecture 1.2 and its restatement, Conjecture 1.6.But is TCC indeed a stronger statement than HC on T in the sense that some hypothesis weaker than TCC implies HC on T ?Our next result confirms this.
We will refer to the statement "∀G ∈ G, χ ′′ (G) ≤ ∆(G) + 3" as weak TCC.In Theorem 3.4, we show that if weak TCC is true for a class F of graphs that is closed under taking subgraphs, then Hadwiger's conjecture is true for the class T (F ).Thus the weaker hypothesis weak TCC implies HC on T ; therefore HC on T is strictly easier than TCC.
We show that for every (∆(G) + 3)-total critical graph G, T (G) has a clique minor of order ∆(G) + 3.
For each fixed positive integer k ≥ 2, let (k)-TCC be the statement, " ∀G ∈ G, χ ′′ (G) ≤ ∆(G)+ k." Thus 2-TCC is same as TCC and 3-TCC is same as weak-TCC.Can we prove that the weaker hypothesis k-TCC (for some fixed positive integer k ≥ 4) implies HC on T ?We do not know the answer, but our approach of showing the existence of a ∆(G) + k clique minor in T (G), assuming that G is (∆ + k)-total critical, does not seem to work when k ≥ 4.More sophisticated approaches may be needed to answer this question.
However, if we have the additional guarantee that the graph G has a high vertex connectivity, then Hadwiger's Conjecture will hold for T (G) if (k)-TCC is true for G.We show in Theorem 4.3 that if (k)-TCC is true, then Hadwiger's conjecture is true for T (F ), where F := {G : κ(G) ≥ 2k − 1} (here, κ(G) denotes the vertex-connectivity of G).By combining this with the upper bound of Molloy and Reed [19], we get that there exists a constant C ′ such that Hadwiger's conjecture is true for T (F ), where Here C ′ = 2C − 1, where C is the constant from Theorem 1.3; in [19] Molloy and Reed proved C = 10 26 , but mention that with more detailed analysis, C can be brought down to 500.

Our contributions to the total coloring literature
In this section we concentrate on the first (and the more straightforward) relaxation of TCC mentioned earlier: weak-TCC.
The first reason that motivated us to go through the known literature on total coloring to find out the important special classes F of graphs for which weak TCC is known to hold (while TCC may still remain unsolved), is the promise of Theorem 3.4 that HC is true for T (F ) for such F .We realized that there are some important graph classes in this category, for example planar graphs.Unfortunately, for general graphs, it seems the chance of either TCC or weak TCC getting proved in the near future is very less.
Is it possible that weak TCC is a more reasonable conjecture than TCC?It is remarkable that another very well-known and widely believed conjecture, the list coloring conjecture, implies the weak TCC, whereas TCC seems beyond its reach.TCC is indeed a bold conjecture, in the sense that at times it makes us think that it is not unreasonable to look for counterexamples.For example, despite intense research for decades, the conjecture could not be proved for planar graphs with maximum degree 6, though it has been found to hold good for all other cases of planar graphs.On the other hand, it is very easy to prove weak TCC for all planar graphs: If G is a planar graph, then χ ′′ (G) ≤ ∆(G) + 3.Moreover, weak TCC is in fact true for a much wider class, namely 4-colorable graphs, which properly includes planar graphs, whereas proving TCC on 4-colorable graphs seems to be extremely difficult.This led us to survey the status of weak TCC on 5-colorable graphs, and try to fill the research gap therein.
Weak TCC can be proved for 4-colorable graphs (that is, when χ(G) ≤ 4) using the following well-known argument: We color the vertices of the graph using the colors {1, 2, 3, 4}.Applying Vizing's theorem we can color the edges of the graph using ∆ + 1 colors from {3, 4, . . ., ∆(G) + 3}.We uncolor all edges with colors 3 or 4. Note that for every uncolored edge, there exist at least two colors in {1, 2, 3, 4} that are not the colors assigned to its endpoints.Let us associate with each uncolored edge the list containing these two colors.It is easy to see that the subgraph induced by the uncolored edges consists only of paths and even cycles, and is therefore 2-edge-choosable.This proves weak TCC for 4-colorable graphs.As a special case, planar graphs satisfy weak TCC since by the four color theorem all planar graphs are 4-colorable.
As mentioned earlier this naturally leads to the question whether weak TCC can be proved for k-colorable graphs with k ≥ 5.It is not the case that researchers who worked on total coloring failed to notice this question altogether.In fact, from the early days of research in total coloring, researchers have tried to find upper bounds for total chromatic number of a graph in terms of its (vertex) chromatic number.
Remark.If we go through the mainstream literature on total coloring, apart from efforts to bring the upper bound closer to the conjectured ∆ + 2, there were efforts to study TCC for graph classes defined by their maximum degree, i.e. ∆ ≤ 4, ∆ ≤ 5 etc. Obviously the other two parameters for such study were the chromatic number χ(G) and the chromatic index χ ′ (G).The latter is essentially ∆, due to Vizing's Theorem.Studying TCC for graphs of bounded chromatic number was the next natural option after bounded maximum degree graphs.
The most important result in this direction was proved by Hind [14].
Theorem 1.9 (Hind).For every graph G, We note that for small values of χ(G) (such as ≤ 9), Theorem 1.9 implies χ ′′ (G) ≤ ∆(G) + 7. Naturally many researchers have attempted to improve this result using adaptations of the technique used in Hind's proof.One such result was presented by Sánchez-Arroyo [23] and a slightly better bound was obtained by Chew [9].
This implies that χ ′′ (G) ≤ χ ′ (G) + 3, for χ(G) ≤ 5.Although weak TCC for class-I 5-colorable graphs follows from this result, weak TCC is not known to hold for the entire class of 5-colorable graphs till date.In Theorem 5.1, we prove the following long-pending result: Weak TCC is true for 5-colorable graphs.
Remark.The method used in our proof is completely different from the proofs of Theorem 1.9 and Theorem 1.10.

Note.
Proving TCC for 5-colorable graphs is likely to be a considerably harder problem.In fact, TCC remains to be proved even for 4-colorable graphs.As mentioned before the ∆ = 6 case for planar graphs is still open even after decades of research.2 Hadwiger's conjecture and total coloring Definition 2.1 (total-critical graph).A graph G is said to be t-total-critical if χ ′′ (G) = t and χ ′′ (G − e) ≤ t − 1, for any edge e of G.
Note that any graph that contains at least one edge has total chromatic number at least 3.
We can assume that for some s < d, s neighbours of v say, v 1 , v 2 , . . ., v s lie in one connected component C 1 , while the remaining d − s neighbours, v s+1 , . . ., v d lie in the remaining connected components C 2 , . . ., C k (k ≥ 2) of H − v. Let the edge vv i be denoted by e i , for all i ≤ d.Define T 1 := H[V (C 1 ) ∪ {v}] and T 2 := H − V (C 1 ).Both T 1 and T 2 can be total colored with (t − 1) colors since H is t-total-critical.
Consider a total coloring ϕ : Note that in ϕ, the colors on the edges e 1 , e 2 , . . ., e s and the vertex v are all different.We can assume without loss of generality (by renaming the colors if necessary) that ϕ(v) = 1 and ϕ(e As in the previous case, in ψ, the colors on the edges e s+1 , e s+2 , . . ., e d and the vertex v are all different.Again by renaming colors if necessary, we can assume that ψ(v) = 1 and ψ(e i ) = i + 1 for each i ∈ {s + 1, s + 2, . . ., d}.Notice that the renaming of colors in both ϕ and ψ is possible because d + 1 ≤ ∆(H) + 1 ≤ t − 1.We now construct a total coloring φ : V (H) ∪ E(H) → [t] of H by combining the colorings ϕ of T 1 and ψ of T 2 (i.e.φ(v) = 1 and for every other element (vertex or edge) x of H, φ(x) = ϕ(x) if x belongs to T 1 and φ(x) = ψ(x) if x belongs to T 2 ).It is easy to verify that φ is a total coloring of H using just t − 1 colors.This contradicts the fact that H is a t-total-critical graph.
Lemma 2.3.Let G be a connected (∆(G) + k)-total-critical graph on at least 3 vertices, for some 2 < k ≤ ∆(G).Then the minimum degree of G is at least k.
Proof.Assume that there exists a vertex w ∈ G with d G (w) ≤ k − 1, and let ∆(G) = ∆.Since G has at least three vertices and there are no cut-vertices in G (Lemma 2.2), w cannot have degree 1.
Let the neighbours of w be v 1 , v 2 , . . ., v j , where 2 ≤ j ≤ k − 1.Call the edge v i w as e i , for each i ∈ [j − 1] and v j w as f .Then G − f has a total coloring using ∆ + k − 1 colors, which we shall assume to be the set Assume that v i is assigned color α i for each i ∈ [j] and the edge e i is assigned color β i for each i ∈ [j − 1].Call the color assigned to w as c. If ), and assign it to the vertex w.Now, the vertices w and v j have different colors on them.Else if α j = c, let c ′ := c.
Since d G (v j ) ≤ ∆, and therefore d G−f (v j ) ≤ ∆ − 1, there exists a list L of at least (k − 1) colors that are not assigned to v j or its incident edges.If at least one of the colors in L is not in {c ′ } ∪ {β i : i ∈ [j − 1]}, assign it to the edge f giving G a (∆ + k − 1)-total coloring.Otherwise since |L| ≥ k − 1 and w has at most k − 2 incident edges in G − f , we have the set of colors that appear on w and its incident edges is exactly the list L; that is, ), and assign it to the vertex w and then assign color c ′ to the edge f , giving a (∆ + k − 1)-total coloring of G.This is a contradiction as χ ′′ (G) = ∆ + k by hypothesis.Hence, d G (w) ≥ k for each vertex w ∈ G.
Theorem 2.4.Let F be a class of graphs that is closed under the operation of taking subgraphs.If TCC is true for F , then Hadwiger's conjecture holds for the class T (F ).
Let H be a (∆ + 2)-total critical subgraph of G.We then have χ ′′ (H) = ∆ + 2. It is clear that ∆(H) = ∆, otherwise since H also belongs to F , TCC would imply that χ ′′ (H . ., e ∆ be the edges incident on v ∆ in H. From Lemma 2.2 we know H − v ∆ is connected.This implies that the subgraph X of T (G) induced by the elements of H − v ∆ is connected.In T (G), contract the subgraph X into a single vertex w.Since the subgraph X contains the v -vertex corresponding to every neighbour of v ∆ , the vertex w is now adjacent to the e-vertices e 1 , . . ., e ∆ and the v -vertex v ∆ , forming a (∆ + 2)-clique minor in T (G).

Weak total coloring conjecture
We may state weaker versions of Conjecture 1.2 by relaxing the upper bound on χ ′′ (G) as follows : Conjecture 3.1 ((k)-TCC ).Let k ≥ 2 be a fixed positive integer.For any graph G, (2)-TCC is the same as Conjecture 1.2 ; that is, the total coloring conjecture.We shall refer to (3)-TCC as the weak total coloring conjecture or weak-TCC for short.
We first make a general observation about graphs that do not contain cut-vertices.
Claim 3.2.Suppose that G is a 2-connected graph and v is a vertex in G. Then there exists a vertex w ∈ N (v) such that {w, v} is not a separator of G.
Proof.Suppose for contradiction that each set in {{v, w}} w∈N (v) is a separator of G. Since G has no cutvertices, we can conclude that each set in {{v, w}} w∈N (v) is a minimum separator of G. Since there is at least one other connected component and it cannot contain x ∈ S, we can conclude that there is a connected component of G − {w, x} that is smaller than S.This contradicts the choice of w and S.
Proof.Recall that if H is not connected, then H contains at most one connected component that is not an isolated vertex.Since the maximum degree of this component is also ∆ and it is also (∆ + 3)-total-critical, we will be done if we prove the statement of the lemma for this component.So from here on, we assume that H is connected.
Given a graph G with a list L e of colors assigned to each edge e, a list edge coloring is a proper coloring of E(G) such that each edge e is assigned a color from the list L e .A graph G is said to be k-edge-choosable if a list edge coloring of G exists for any assignment of lists of size k to the edges of G.The smallest positive integer k such that G is k-edge-choosable is called the edge choosability of G, which is denoted as ch ′ (G).The following is a well-known fact.
The list coloring conjecture is as follows : Theorem 3.8.Let F be a class of graphs closed under the operation of taking subgraphs.Hadwiger's conjecture holds for the class T (F ), if F satisfies the list coloring conjecture.
Proof.Since F satisfies the list coloring conjecture, for each Lemma 3.6 and Theorem 1.1), that is, F satisfies weak TCC.The theorem then follows from Theorem 3.4.
Thus, the validity of the list coloring conjecture would imply that Hadwiger's conjecture is true for all total graphs.

Connectivity and Hadwiger's conjecture for total graphs
Given a partition of the vertex set of a graph, an edge of the graph is said to be a cross-edge if its endpoints belong to different sets of the partition.
Our next result makes use of the following well-known theorem : Theorem 4.1 (Tutte; Nash-Williams; See [11]).A multigraph contains k edge-disjoint spanning trees if and only if every partition P of its vertex set contains at least k(|P | − 1) cross-edges.
Proof.Let v ∆ be a vertex of maximum degree in G. Let the set of e-vertices in T (G) corresponding to the edges in E G (v ∆ ) be denoted by Z.Note that Z induces a ∆-clique (call it K Z ) in T (G).Let H be the graph obtained from G by removing v ∆ .We will show that there exist k disjoint but pair-wise adjacent connected induced subgraphs of T (H), so that by contracting these induced subgraphs we get k more vertices that can be added to the existing ∆-clique K Z thus achieving the desired (∆ + k)-clique minor in T (G).Since G is (2k − 1)-connected, the vertex-connectivity of H is at least 2k − 2. It follows that, edgeconnectivity of H is also at least 2k − 2. Hence, by Corollary 4.2, H has (k − 1) pair-wise edge-disjoint spanning trees.
Let T 1 , . . ., T k−1 be the pair-wise edge-disjoint spanning trees in H. Let E ′ i be the set of e-vertices in T (G) corresponding to the edges in E(T i ), for i ∈ [k − 1].Clearly the induced subgraph of T (G) on E ′ i is isomorphic to the line graph of T i and is connected and therefore, can be contracted to a single vertex, say y i .Let Y := {y i : i ∈ [k − 1]}.The set of v -vertices of T (G) is also connected since they induce a subgraph isomorphic to H itself, and can be contracted to a single vertex v * .Let F be the graph so obtained.Since T 1 , . . ., T k−1 are all spanning trees of H, it follows that y i y j ∈ E(F ) for distinct i, j ∈ [k − 1] and also that v * y i ∈ E(F ) for all i ∈ [k − 1].Further, it can be seen that in F , each y i ∈ Y and v * are adjacent to all the e-vertices in Z.Therefore, Y ∪ Z ∪ {v * } forms a clique of order ∆ + k in F , implying that T (G) contains a (∆ + k)-clique minor.
Our goal now is to start with the coloring π of G and then recolor each conflicting edge so that it is no longer a conflicting edge in the resulting coloring.We will do so in three phases and in each phase, we select a color i ∈ {3, 4, 5} and recolor the conflicting i-edges one by one.Our recoloring strategy will rely on the following lemma.(f ) if e ′ is a conflicting i-edge in G, then it has original color i, that is, π(e ′ ) = ϕ(e ′ ) = i.
Further, let e be a conflicting i-edge in G. Then G can be recolored to a coloring ψ so that e is no longer a conflicting edge in ψ, and the conditions (a) -(f ) hold for this new coloring ψ too.
Proof.Let e = v 0 v 1 , where ϕ(e) = ϕ(v 1 ) = i.By (f), we know that the original color of e is i.
If there exists a color in A i \ (S ϕ (v 0 ) ∪ S ϕ (v 1 )), we can recolor e with that color to get ψ.It is easy to verify that all the properties (a) -(f) hold for ψ.
So we shall assume that A i ⊆ S ϕ (v 0 ) ∪ S ϕ (v 1 ).Let X be the set of edges incident on v 0 that have original color in [3, i].Since |[3, i]| = i − 2 and π| E(G) is a proper edge coloring, we have |X| ≤ i − 2. Since v 0 v 1 has original color i, we have v 0 v 1 ∈ X, which implies that |X \ {v 0 v 1 }| ≤ i − 3. Since ϕ(v 0 v 1 ) = i, it is not an A i -edge.Thus there are at most i − 3 A i -edges in X.Since every A i -edge has original color in [3, i] by (d), we know that every A i -edge incident on v 0 is in X.It follows that there are at most i − 3 A i -edges incident on v 0 .Considering the possibility that ϕ(v 0 ) also may be in A i , we have |S ϕ (v 0 ) ∩ A i | ≤ i − 2. This implies that there exists a color γ ∈ A i \ S ϕ (v 0 ).As A i ⊆ S ϕ (v 0 ) ∪ S ϕ (v 1 ), we have γ ∈ S ϕ (v 1 ).Therefore, as ϕ(v 1 ) = i, there exists an edge v 1 v 2 such that ϕ(v 1 v 2 ) = γ.As γ / ∈ S ϕ (v 0 ), we have v 1 v 2 = v 0 v 1 .We call an edge f in G an i-edge if ϕ(f ) ∈ A i \ {γ} and the original color of f is i.By (e), we know that one of the endpoints of every i-edge is colored i.Let Γ and I be the set of all γ-and i-edges in G, respectively.Now consider a maximal (Γ, I)-trail P = v 1 v 2 . . .v k , where k ≥ 2, starting with the edge v 1 v 2 .It is easy to see that every edge on P is an A i -edge.Then we have the following: (1) For each odd t ∈ {1, 2, . . ., k}, ϕ(v t ) = i.
(2) v 0 P is a path.
Otherwise, some vertex must be repeated while traversing v 0 P .Let v be the first such vertex.If v = v 0 , then it either has two edges having original color i or two γ-edges incident on it; as both ϕ| E(G) and π| E(G) are proper edge colorings of G, this is a contradiction.Now, let v = v 0 .Recall that by choice of γ, the vertex v 0 has no incident γ-edge.Then, since v 0 v 1 has original color i, there must be two incident edges with original color i at v 0 ; another contradiction.It follows that v 0 P is a path.
Note that the last vertex v k on P satisfies the following properties: To see this, note that in the first case, we have that k is odd and then by (1), ϕ(v k ) = i = γ.By the maximality of P , we have that v k has no incident γ-edges.In the second case, suppose A i ⊆ S ϕ (v k ).This happens only if, for each c ∈ A i \ {ϕ(v k )}, v k has an c-edge incident on it.By property (d) and the fact that π| E(G) is a proper edge coloring, these i − 2 edges have a distinct original color from the set [3, i].Since |[3, i]| = i − 2, we can conclude that v k has an incident edge v k w with original color i and ϕ(v k w) ∈ A i .Is it possible that w = v k−1 ?Recall that in this case, k − 1 is odd and v k−2 v k−1 is an i-edge (when k = 2) or an i-edge (when k > 2).As π| E(G) is a proper edge-coloring, v k−1 v k cannot have original color i and so w = v k−1 .As ϕ(v k−1 v k ) = γ, we have ϕ(v k w) = γ, and so v k w is an i-edge.This means v k has an incident i-edge outside of P .This violates the maximality of the trail P .So there exists a color c ′ ∈ A i \ S ϕ (v k ).
It is easy to see that properties (a), (d), (e), and (f) hold for this new coloring ψ: We have not recolored any vertex, therefore (a) holds; Other than v 0 v 1 , all the edges that we have recolored are A i -edges (with respect to ϕ) and they remain A i -edges with respect to ψ as well; The only new A i -edge with respect to ψ is e = v 0 v 1 .This is consistent with (d) and (e); the set of conflicting i-edges with respect to ψ is the set of conflicting i-edges with respect to ϕ minus v 0 v 1 , therefore (f) holds.We shall now proceed to show that the color shift strategy produces no new conflicts, ensuring that ψ satisfies properties (b) and (c) also.
Since we are recoloring only the edges on v 0 P , if (b) or (c) is violated in ψ, there must be an edge on v 0 P such that it conflicts either with one of its end-points or with an adjacent edge.Let t be any index such that v t v t+1 is such an edge with respect to ψ.Let c = ψ(v t v t+1 ).We will say that the edge v t v t+1 has a conflict at the end-point v t+1 , if either ψ(v t+1 ) = c or v t+1 has an incident edge v t+1 u = v t v t+1 with ψ(v t+1 u) = c.Otherwise, the conflict has to be at the end-point v t .
We claim that v t v t+1 cannot have a conflict at the end-point v t+1 .If t = k − 1, this is obvious from (3a) and (3b).For t ≤ k − 2, ψ(v t v t+1 ) = ϕ(v t+1 v t+2 ) = c.Suppose that ψ(v t+1 ) = c.Then since we are recoloring only edges, we have ϕ(v t+1 ) = c, which contradicts the fact that ϕ satisfies (c).Next, suppose that there exists an edge v t+1 u = v t v t+1 such that ψ(v t+1 u) = c.Since no two consecutive edges of v 0 P have the same color in ψ, we know that u = v t+2 .Then v t+1 u is not an edge of v 0 P which means that ϕ(v t+1 u) = ψ(v t+1 u) = c, which implies that there are two edges with color c incident at v t+1 in ϕ.This contradicts the fact that ϕ satisfied (b).Therefore, in ψ, the edge v t v t+1 cannot have a conflict at the end-point v t+1 .
Hence, v t v t+1 has a conflict at the end-point v t .We consider the two possible cases separately.
In this case the edge v t v t+1 is an i-edge (if t > 0) or i-edge (if t = 0) before the recoloring.Note that this means that t + 1 is odd.If t = 0, then by the choice of γ, we have γ / ∈ S ϕ (v 0 ) and thus the edge v 0 v 1 has no conflict at the end-point v 0 in ψ.For t ≥ 2, since ϕ(v t−1 v t ) = γ, and ϕ satisfied (c) and (b), we know that ψ(v t ) = ϕ(v t ) = γ and that no edge incident at v t other than v t−1 v t , v t v t+1 has the color γ in ψ.Since in ψ, no two consecutive edges on v 0 P have the same color, ψ(v t−1 v t ) = γ, and so we can conclude that v t v t+1 has no conflict at the end-point v t in ψ in this case.
5. In the proof of Theorem 5.1, χ ′ (G) colors are used to do the initial proper edge coloring and an additional two colors are used to get a proper vertex coloring.This suggests the question whether it is possible, when χ ′ (G) = ∆(G) (that is, when G is class I ), to obtain the bound χ ′′ (G) ≤ ∆(G) + 2, instead of the ∆(G)+ 3 bound that we proved in Theorem 5.1.Unfortunately, our proof fails to achieve this since even when dealing with class I graphs, we need one extra color, the (∆ + 3)-th color, to establish the Property A for the original coloring.Property A is crucial in phase III of the recoloring process, that is, when conflicting 5-edges are recolored (see proof of Lemma 5.2).
It would be interesting to prove, χ ′′ (G) ≤ ∆(G) + 2 when G is a class I 5-colorable graph, possibly by some clever tweaking of the proof of Theorem 5.1.

Lemma
For w ∈ N (v), let F (v, w) be the number of vertices in the smallest component in G − {v, w}.Choose w ∈ N (v) such that F (v, w) is as small as possible.Let S be the smallest component in G − {v, w}.Since {v, w} is a minimum separator, v has a neighbour in each component of G − {v, w}, and so also in S. Let x ∈ S ∩ N (v).Now consider the graph G − {w, x}.Since w had a neighbour in each connected component of G − {v, w} (as {v, w} is a minimum separator), all the components of G − {v, w}, together with w, now form one connected component of G − {w, x}.Thus the other connected components of G − {w, x} must all be subgraphs of S.

Lemma 5 . 2 .
Let ϕ be a coloring of V (G)∪E(G) at a certain stage of the recoloring process.Let i ∈ {3, 4, 5} and let A i = {1, 2, . . ., i − 1}.Suppose ϕ satisfies the following properties:(a) ϕ| V (G) = π| V (G) , (b) ϕ| E(G) is a proper edge coloring of G, (c) there are no conflicting A i -edges in G, (d) every A i -edge in G has original color in {3, 4, . . ., i},(e) every A i -edge in G that has original color i has an endpoint colored i, that is, it was originally a conflicting i-edge, and [13]n an edge e = uv in G, the contraction of the edge e involves the following : deleting vertices u and v, introducing a new vertex w e , and making w e adjacent to all vertices in the set(N G (u) ∪ N G (v)) \ {u, v}.The new graph thus obtained is denoted by G/e.A graph H is said to be a minor of G if a graph isomorphic to H can be obtained from G, by performing a sequence of operations involving only vertex deletions, edge deletions, and edge contractions.If G contains H as a minor, then we write H G.The celebrated Hadwiger's conjecture is a far-reaching generalization of the Four Color Theorem.It was proposed by Hugo Hadwiger[13]in 1943.
2.2.If a connected graph H is t-total-critical, where t ≥ ∆(H) + 2, then H has no cut-vertices.Proof.Let H be a connected t-total-critical graph, where t ≥ ∆(H) + 2. Let us choose a vertex v in H with d H