Coloring (gem, co-gem)-free graphs
Journal of Graph Theory
A gem is a graph that consists of a path on four vertices plus a vertex adjacent to all four vertices of the path. A co-gem is the complement of a gem. We prove that every (gem, co-gem)-free graph G satisfies the inequality ��χ(G��) ≤ 5ω(G)/4 (a special case of a conjecture of Gyárfás) and the inequality ��χ(G��) ≤ Δ(G)+ω(G)+1/2(a special case of a conjecture of Reed). Moreover, we give an O(n3) -time algorithm that computes the chromatic number of any (gem, co-gem)-free graph with n vertices, while the existing algorithm in the literature takes O(n��217+1).
Karthick, T. and Maffray, Frédéric, "Coloring (gem, co-gem)-free graphs" (2018). Journal Articles. 1180.