The game chromatic number of some join graphs (Q2786882)

Let \(G\) be an undirected graph. The graph \(k\)-coloring game on \(G\) is played by two players, Alice and Bob, who alternatively properly color an uncolored vertex of \(G\) with a color from \(\{1,\dots,k\}\). Alice wins the game if and only if the graph \(G\) is eventually properly colored. The smallest \(k\) for which Alice has a winning strategy for the graph \(k\)-coloring game is the game chromatic number of \(G\).NEWLINENEWLINEThe join graph \(G\vee H\) is the graph obtained from graphs \(G\) and \(H\) by adding edges between every vertex of \(G\) and every vertex of \(H\). In this paper, the authors determine the value of the game chromatic number of join graphs \(P_n\vee K_m\), \(n,m\geq 1\), which appears to be not so difficult.