(2P_2, K_4)-FREE GRAPHS ARE 4-COLORABLE

摘要

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 [J. Combin. Theory Ser. B, 29 (1980), pp. 345{346] from 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 [Discrete Appl. Math., 216 (2017), pp. 273{280] 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.