The size of edge chromatic critical graphs with maximum degree 6

Abstract

It is worth to mentioning that Chetwynd and Hilton [5] showed that the above conjecture is not true for multigraphs.Jakobsen [8] showed that if G is 3-critical, then m ≥ 4 3 n, thus verifying the conjecture when = 3. Fiorini and Wilson [7] showed that if G is a -critical Journal of Graph Theory

The average degree of an edge‐chromatic critical graph II

“…Lemma 2.4 implies the results in ( [5], Lemma 2.3) and ( [3], Lemma 2.3), and its proof is based on the proofs of those results; see the Acknowledgments at the end of this article.…”

“…I am grateful to Professors R. Luo and Y. Zhao for sending me a pre-publication copy of reference [3]. Lemma 2.3 of [3] is the special case of Corollary 2.6 in which d(x) = 3; its proof was the inspiration for Lemma 2.4, without which this article could not have been written.…”

“…Lemma 2.3 of [3] is the special case of Corollary 2.6 in which d(x) = 3; its proof was the inspiration for Lemma 2.4, without which this article could not have been written. While this article was in preparation, Luo and Zhao [4] independently obtained a weaker version of its results, as mentioned before Theorem 1.1.…”

“…Using an appropriate subset of these lemmas, it is proved in [12] thatd Corollary 2.5 is very similar to ([5], Lemma 2.3), which also follows from Lemma 2.4, but which we do not need in the present article. In fact, the proof of Theorem 1.1 (a), and of the truth of (b) and (c) for sufficiently large , needs only Corollary 2.6, which is a generalization of ( [3], Lemma 2.3). It is only in modifying the proof in order to prove (b) and (c) for smaller values of that we have used Corollary 2.5.…”

Finding the exact bound of the maximum degrees of class two graphs embeddable in a surface of characteristic ϵ∈{−1,−2,−3}

“…The proofs of next two lemmas can be found in [10]. Before we state our next lemma, we introduce the following notation.…”