Structure of some ( P 7, C 4)-free graphs with application to colorings

We use P t and C t to denote a path and a cycle on t vertices, respectively. A diamond is a graph obtained from two triangles that share exactly one edge. A kite is a graph that consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A gem is a graph that consists of a P 4 plus a vertex adjacent to all vertices of the P 4. Choudum et al. (2021) proved that every (P 7, C 7, C 4, gem)-free graph has either a clique cutset or a vertex whose neighbors is the union of two cliques unless it is a clique blowup of the Petersen graph, and every non-Petersen (P 7, C 7, C 4, diamond)-free graph has either a clique cutset or has a vertex of degree at most max { 2, ω ( G ) − 1 }. In this paper, we extend these two results respectively to (P 7, C 4, gem)-free graphs and (P 7, C 4, diamond)-free graphs. We also prove that if G is a (P 7, C 4, kite)-free graph with δ ( G ) ≤ ω ( G ) and without clique cutsets, then there is an integer ℓ ≥ 0 such that G = K ℓ + H, where H is either the Petersen graph or a well-defined graph F. This implies a polynomial algorithm which can determine the chromatic number of (P 7, C 4, diamond)-free graphs, color (P 7, C 4, kite)-free graphs G with ( ω ( G ) + 1 ) colors, and color (P 7, C 4, gem)-free graphs G with ( 2 ω ( G ) − 1 ) colors.