On minimal triangle-free planar graphs with prescribed 1-defective chromatic number

TL;DR

This paper determines the smallest triangle-free planar graph with a 1-defective chromatic number of 3, establishing that it has 11 vertices, contributing to the understanding of defective colorings in planar graphs.

Contribution

The paper precisely identifies the minimal order of triangle-free planar graphs with a 1-defective chromatic number of 3 as 11 vertices, a new exact result in graph coloring theory.

Findings

f(3,1;tfp)=11, the minimal order of such graphs

Established the exact size of minimal graphs with given defective chromatic number

Contributed to the classification of triangle-free planar graphs by their defective chromatic properties

Abstract

A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on the set of all vertices receiving the same colour is at most k. The k-defective chromatic number $\chi_k(G)$ is the least positive integer $m$ for which graph G is (m,k)-colourable. Let f(m,k;tfp) be the smallest order of a triangle-free planar graph such that $\chi_k(G)$=m. In this paper we show that f(3,1;tfp)=11.