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arXiv · 1807.05547

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

Serge Gaspers·Shenwei Huang
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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.

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Serge Gaspers, Shenwei Huang. 2018-12-14. $(2P_2,K_4)$-Free Graphs are 4-Colorable. https://arxiv.org/abs/1807.05547

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