Hadwiger's Conjecture for 3-Arc Graphs

Wood, David R.; Xu, Guangjun; Zhou, Sanming

doi:10.37236/5134

Cited by 6 publications

(5 citation statements)

References 22 publications

(35 reference statements)

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“…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].…”

Section: Theorem 9 ([24]mentioning

confidence: 84%

Weak Total Coloring Conjecture and Hadwiger’s Conjecture on Total Graphs

Basavaraju,

Chandran,

Francis

et al. 2024

Electron. J. Combin.

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The total graph of a graph $G$, 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 $G$. The total chromatic number $\chi''(G)$ of a graph $G$ is defined to be the chromatic number of its total graph. The well-known Total Coloring Conjecture or TCC states that for every simple finite graph $G$ having maximum degree$\Delta(G)$, $\chi''(G)\leq \Delta(G) + 2$. In this paper, we consider two ways to weaken TCC: (1) Weak TCC: This conjecture states that for a simple finite graph $G$, $\chi''(G) = \chi(T(G)) \leq\Delta(G) + 3$. While weak TCC is known to be true for 4-colorable graphs, it has remained open for 5-colorable graphs. In this paper, we settle this long pending case. (2) Hadwiger's Conjecture for total graphs: We can restate TCC as a conjecture that proposes the existence of a strong $\chi$-bounding function for the class of total graphs in the following way: If $H$ is the total graph of a simple finite graph, then $\chi(H) \leq\omega(H) + 1$, where $\omega(H)$ is the clique number of $H$. A natural way to relax this question is to replace $\omega(H)$ by the Hadwiger number $\eta(H)$, the number of vertices in the largest clique minor of $H$. This leads to the Hadwiger's Conjecture (HC) for total graphs: if $H$ is a total graph then $\chi(H) \leq \eta(H)$. We prove that this is true if $H$ is the total graph of a graph with sufficiently large connectivity. A second motivation for studying Hadwiger's conjecture for total graphs is the following: Consider the class of split graphs whose vertex set is partitioned into an independent set $A$ and a clique $B$, with the following additionalconstraints: (1) Each vertex in $B$ has exactly 2 neighbours in $A$; (2) No two vertices in $B$ have the same neighbourhood in $A$. It is known that (European Journal of Combinatorics, 76, 159-174,2019) if Hadwiger's conjecture is proved for the squares of this special class of split graphs, then it holds also for the general case. Of course, proving the conjecture for this specialzed-looking case is indeed difficult since it is only a reformulation of the general case, and therefore it is natural to consider the difficulty level of Hadwiger's conjecture for the squares of graph classes defined by slighly modifying the above class of graphs. A natural structural modification is to assume that both $A$ and $B$ are independent sets, keeping everything else same. It turns out that the squares of this modified class of graphs is exactly the class of total graphs. From this perspective, it is not really surprisingthat HC on Total Graphs is also challenging. On the other hand, we show that weak TCC implies HC on total graphs. This perhaps suggests that the latter is an easier problem than the former.

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Section: Theorem 9 ([24]mentioning

confidence: 84%

Weak Total Coloring Conjecture and Hadwiger’s Conjecture on Total Graphs

Basavaraju,

Chandran,

Francis

et al. 2024

Electron. J. Combin.

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“…3 ) \ C(14, 3) is simply ( [13] 3 ), by Table I and the proof of Lemma 12 we know that ( [14] 3 ) \ C(14, 3) contains a complete minor of order at least f (13) = 88 such that the hyperedges (of K 3 14 ) in each of its branch sets cover at least 13/2 = 7 labels of [13].…”

Section: Proof Since ( [14]mentioning

confidence: 95%

“…It has been proved for graphs with

χ (G) \leq 6

, and is open for graphs with

χ (G) \geq 7

. It has also been proved for certain special classes of graphs, including powers of cycles and their complements , proper circular arc graphs , line graphs , quasiline graphs , and 3‐arc graphs . See for a survey.…”

Section: Introductionmentioning

confidence: 99%

Hadwiger's Conjecture for the Complements of Kneser Graphs

Zhou

2015

J. Graph Theory

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“…Hadwiger's conjecture has been proved for several classes of graphs, including line graphs [13], proper circular arc graphs [3], quasi-line graphs [6], 3-arc graphs [16], complements of Kneser graphs [17], and powers of cycles and their complements [10]. Since Hadwiger's conjecture is equivalent to the statement that η(G) ≥ χ(G) for any graph G, where χ(G) is the chromatic number of G, there is also an extensive body of work on the Hadwiger number.…”

Section: Introductionmentioning

confidence: 99%

Hadwiger’s Conjecture and Squares of Chordal Graphs

Chandran

Issac

Zhou

2016

Lecture Notes in Computer Science

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Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split graphs. Since all split graphs are chordal, this implies that Hadwiger's conjecture is true for all graphs if and only if it is true for squares of chordal graphs. It is known that 2-trees are a class of chordal graphs. We further prove that Hadwiger's conjecture is true for squares of 2-trees. In fact, we prove the following stronger result: for any 2-tree T , its square T 2 has a clique minor of order χ(T 2 ) for which each branch set is a path, where χ(T 2 ) is the chromatic number of T 2 . As a corollary, we obtain that the same statement holds for squares of generalized 2-trees, and so Hadwiger's conjecture is true for such squares, where a generalized 2-tree is a graph constructed recursively by introducing a new vertex and making it adjacent to a single existing vertex or two adjacent existing vertices in each step, beginning with the complete graph of two vertices.

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Hadwiger's Conjecture for 3-Arc Graphs

Abstract: The 3-arc graph of a digraph D is defined to have vertices the arcs of D such that two arcs uv, xy are adjacent if and only if uv and xy are distinct arcs of D with v = x, y = u and u, x adjacent. We prove that Hadwiger's conjecture holds for 3-arc graphs.

Cited by 6 publications

References 22 publications

Weak Total Coloring Conjecture and Hadwiger’s Conjecture on Total Graphs

Weak Total Coloring Conjecture and Hadwiger’s Conjecture on Total Graphs

Hadwiger's Conjecture for the Complements of Kneser Graphs

Hadwiger’s Conjecture and Squares of Chordal Graphs

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