$(2P_2,K_4)$-Free Graphs are 4-Colorable

TL;DR

This paper proves that all $(2P_2,K_4)$-free graphs are 4-colorable, resolving an open question from the 1980s, and provides a 2-approximation algorithm for coloring $(4P_1,C_4)$-free graphs.

Contribution

It establishes the 4-colorability of $(2P_2,K_4)$-free graphs and offers the first positive approximation result for coloring $(4P_1,C_4)$-free graphs.

Findings

$(2P_2,K_4)$-free graphs are 4-colorable

Bound is tight, attained by specific graphs

Provides a 2-approximation algorithm for $(4P_1,C_4)$-free graphs

Abstract

In this paper, we show that every $(2P_2,K_4)$-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon \cite{Wa80} in the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for $(4P_1,C_4)$-free graphs. Lozin and Malyshev \cite{LM17} conjecture that the answer is yes. As an application of our main result, we provide the first positive evidence to the conjecture by giving a 2-approximation algorithm for coloring $(4P_1,C_4)$-free graphs.