Some completion problems for graphs without chordless cycles of prescribed lengths

Given a proper vertex coloring of a graph G, a supergraph that remains properly colored is called a completion of G. In this paper, we first consider the problem of determining whether a given properly colored graph has a C4-free completion, and completely classify the complexity of this problem based on the number of colors used on the input graph. We show that this problem is polynomial time solvable when three colors are used, but NP-complete when the graph is colored with k≥4 colors. We then show that an identical dichotomy exists for the problem of completing a colored graph to be (C4,C5)-free. In contrast, we also show that for any fixed s≥1 the problem of determining if a k-colored graph has a (C4,C5,…,Cs+5)-free completion and the problem of determining if a k-colored graph can be completed to be (C4,C6,…,C2s+4)-free are NP-complete when the input graph is k-colored for any fixed k, k≥3.