Balanced Cayley graphs and balanced planar graphs

A balanced graph is a bipartite graph with no induced circuit of length 2(mod4). These graphs arise in integer linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented. The graphs in this paper are simple.