The chromatic index of graphs with large maximum degree

Chromatic-Index-Critical Graphs of Orders 11 and 12

“…For k = 3 and 4, this is proved in [1] and [9,11], respectively. For k = 9, this follows from more general results of [6], and for k = 10, 11, it is a consequence of the results of [17,18].…”

Graphs which are vertex‐critical with respect to the edge‐chromatic class

“…The conclusion of Conjecture 2 has been verified in a number of special cases ( [1], [2], [3], [4], [5], [12], [13]), and, in particular, when A(G) s= | V(G)\ -3. We consider below a number of conjectures which are in fact implied by Conjecture 2; all of these conjectures are also verified when…”

Overfull conjecture for graphs with high minimum degree

“…The result follows immediately from Corollary 2.1 when ! ð ffiffi ffi 7 p =3Þn because Á < n. To improve this to the stated result, we first note that the Overfull Conjecture is clearly true when n is even and Á ¼ n À 1, because 0 ðK n Þ ¼ n À 1: For Á ¼ n À 2, the Overfull Conjecture was verified in [14]. Therefore, we may assume that Á n À 3: Substituting this value for Á into Corollary 2.1 then yields the result.…”

“…The Overfull Conjecture has been directly verified in a few very restrictive cases for given maximum degree when G is not regular (see, e.g., Plantholt [13,14]), but no result comparable to Theorem 1.1 has been achieved previously. Other researchers have used more complicated versions of the arguments in this paper to achieve conditions that guarantee a graph is in Class 1.…”