Graphs with Odd Cycle Lengths 5 and 7 are 3-Colorable

Let L(G) denote the set of all odd cycle lengths of a graph G. Gyárfás gave an upper bound for χ(G) depending on the size of this set: if |L(G)| = k ≥ 1, then χ(G) ≤ 2k+1 unless some block of G is a K2k+2, in which case χ(G) = 2k+2. This bound is generally tight, but when investigating L(G) of special forms, better results can be obtained. Wang completely analyzed the case L(G) = {3, 5}; Camacho proved that if L(G) = {k, k+2}, k ≥ 5, then χ(G) ≤ 4. We show that L(G) = {5, 7} implies χ(G) = 3.