  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.
