On graphs double-critical with respect to the colouring number

TL;DR

This paper characterizes double-col-critical graphs with coloring number up to 5, analyzes their properties, and explores the ratio of double-col-critical edges in k-critical graphs for k > 4.

Contribution

It provides a complete characterization of double-col-critical graphs with coloring number at most 5 and investigates the proportion of double-col-critical edges in k-critical graphs.

Findings

Double-col-critical graphs with col(G) ≤ 5 are characterized.

In 4-col-critical non-complete graphs, at most half of the edges are double-col-critical.

Extremal graphs with maximum double-col-critical edges are odd wheels on at least six vertices.

Abstract

The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph is removed at any step. An edge e of a graph G is said to be double-col-critical if the colouring number of G-V(e) is at most the colouring number of G minus 2. A connected graph G is said to be double-col-critical if each edge of G is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer k greater than 4 and any positive number r, there is a k-col-critical graph with the ratio of double-col-critical edges between 1- r and 1.