On triangle-free graphs of order 10 with prescribed 1-defective chromatic number

TL;DR

This paper investigates the structure of triangle-free graphs of order 10 with a 1-defective chromatic number of 3, identifying a unique critical graph and showing the prevalence of certain subgraphs.

Contribution

It characterizes the unique (3,1)-critical triangle-free graph on 10 vertices and demonstrates the common presence of specific subgraphs in larger graphs with the same chromatic number.

Findings

Four triangle-free graphs of order 9 have 1-defective chromatic number 3.

Almost all such graphs of order 10 contain these four as subgraphs.

There is a unique (3,1)-critical triangle-free graph on 10 vertices.

Abstract

A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least positive integer m for which the graph is (m, k)-colourable. All triangle-free graphs on 8 or fewer vertices are (2, 1)-colourable. There are exactly four triangle-free graphs of order 9 which have 1-defective chromatic number 3. We show that these four graphs appear as subgraphs in almost all triangle-free graphs of order 10 with 1-defective chromatic number equal to 3. In fact there is a unique triangle-free (3, 1)-critical graph on 10 vertices and we exhibit this graph.