Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions

Abstract

Let G be a simple graph, and let ∆(G) and χ ′ (G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ ′ (G) = ∆(G) or ∆(G) + 1. We say G is ∆-critical if χ ′ (G) = ∆ + 1 and χ ′ (H) < χ ′ (G) for every proper subgraph H of G. In 1968, Vizing conjectured that if G is a ∆-critical graph, then G has a 2-factor. Let G be an n-vertex ∆-critical graph. It was proved that if ∆(G) ≥ n/2, then G has a 2-factor; and that if ∆(G) ≥ 2n/3 + 12, then G has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a ∆-critical graph under "moderate" given toughness and maximum degree conditions. In particular, we show that if G is an n-vertex ∆-critical graph with toughness at least 3/2 and with maximum degree at least n/3, then G has a 2-factor. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs

“…Let G be a graph without a 2-factor and (S, T ) be a barrier of G. For an integer k ≥ 0, let C 2k+1 denote the set of odd components D of G − (S ∪ T ) such that e G (D, T ) = 2k + 1. The following result was proved as a claim in [4] but we include a short proof here for self-completeness.…”

“…Let y 2 ∈ T such that e G (x 2 , T ) = e G (x 2 , y 2 ) = 1. The vertex y 2 uniquely exists by the choice x 2 and Lemma 8 (4). By Lemma 8( 1) and ( 4), and the choice of P , we know that y 1 x 1 P x 2 y 2 and T \ {y 1 , y 2 } together contains an induced P 4 ∪ aP 1 .…”

“…By Lemma 7, if G does not have a 2-factor, then G has a barrier. In [4], a biased barrier of G is defined as a barrier (S, T ) of G such that among all the barriers of G,…”

Forbidden subgraphs and 2-factors in 3/2-tough graphs

“…Then by Theorem 2.1, there exists a pair (S, T ) of disjoint subsets of V (G) with η(S, T ) ≤ −2, and we call such a pair a Tutte pair for G. We define that a Tutte pair (S, T ) is called a special Tutte pair if among all the Tutte pairs for G, We use the following lemma for a special Tutte pair. This lemma was proved, for example, in [12].…”

On the chromatic edge stability index of graphs

“…Vizing later proposed two conjectures for critical Class 2 graphs, notably, in [18] he conjectured that every such graph admits a 2-factor, and in [19] that the independence number of a critical graph is at most half of its order. In spite of many attempts and partial results, the conjectures are still unresolved; see a recent study [11] and the references therein.…”