Discrete Mathematics
Plane elementary bipartite graphs
Discrete Applied Mathematics
Planar k-cycle resonant graphs with k = 1,2
Discrete Applied Mathematics
Graph Theory
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The permanental polynomial of a graph G is @p(G,x)@?per(xI-A(G)). From the result that a bipartite graph G admits an orientation G^e such that every cycle is oddly oriented if and only if it contains no even subdivision of K"2","3, Yan and Zhang showed that the permanental polynomial of such a bipartite graph G can be expressed as the characteristic polynomial of the skew adjacency matrix A(G^e). In this note we first prove that this equality holds only if the bipartite graph G contains no even subdivision of K"2","3. Then we prove that such bipartite graphs are planar. Unexpectedly, we mainly show that a 2-connected bipartite graph contains no even subdivision of K"2","3 if and only if it is planar 1-cycle resonant. This implies that each cycle is oddly oriented in any Pfaffian orientation of a 2-connected bipartite graph containing no even subdivision of K"2","3. Accordingly, we give a way to compute the permanental polynomials of such graphs by Pfaffian orientation.