A Characterization on Pfaffian Graphs of Cartesian Product G × C 2 n + 1

The recognition of Pfaffian bipartite graphs in polynomial time has been obtained, but this fact is still unknown for Pfaffian nonbipartite graphs. For a path $P_n$ and a cycle $C_n$, the Pfaffian graphs of Cartesian products $G\times P_{2n}$ and $G\times C_{2n}$ were characterized by Lu and Zhang in 2014. Recently, Li and Wang characterized the Pfaffian graph $G\times P_{2n+1}$ for $n\ge 2$ and the Pfaffian graph $G\times P_3$ for $G$ is bipartite. However, the question of characterizing the Pfaffian graph $G\times C_{2n+1}$ is still open. In this paper, we try to investigate this question for the graph $G$ with a perfect matching. We first prove that $G\times C_{2n+1}$ ($n\ge 3$) and $G\times C_5$ are Pfaffian if and only if $G$ is an odd path or an even cycle, respectively. After showing that $G\times C_3$ is not Pfaffian if G is nonbipartite, we obtain the characterization of the Pfaffian graph $G\times C_3$ in terms of the forbidden subgraphs of $G$.