Chromatic-Index-Critical Graphs of Orders 11 and 12

Abstract

A chromatic-index-critical graph G on n vertices is non-trivial if it has at most n 2 edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic-index-critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial chromatic-index-critical graphs of order ≤ 12.

Counting Finite Lattices

“…[1,3,4,6,11,13,15,16,17]). In this paper, we introduce a fast orderly algorithm that generates lattices, i.e.…”

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

“…Let x be a j -vertex that is adjacent to a k-vertex y. If j < Δ, k < Δ, then x is adjacent to at least Δ − k + 1 vertices z satisfying the following: z = y; z is adjacent to at least 2Δ − j − k vertices different from x of degree at least 2Δ − j − k + 2; and if z is not adjacent to y, then z is adjacent to at least The next lemma summarizes the results in [3][4][5][6][7]. Proof.…”

Independent sets and 2‐factors in edge‐chromatic‐critical graphs

“…Thus it is not the specific structure of the graph which causes its coloring properties. The same holds true for many critical graphs with an odd number of vertices and Áb jVj 2 c þ 1 edges, and it seems that for a fixed odd number n the most critical graphs of order n are overfull, see [1].…”