Walk-powers and homomorphism bound of planar graphs

As an extension of the Four-Color Theorem it is conjectured that every planar graph of odd-girth at least 2k + 1 admits a homomorphism to PC2k = (Z 2 , {e1, e2, · · · , e2k, J}) where ei’s are standard basis and J is all 1 vector. Noting that PC2k itself is of odd-girth 2k + 1, in this work we show that if the conjecture is true, then PC2k is an optimal such a graph both with respect to number of vertices and number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for bipartite signed planar graphs of unbalanced-girth 2k. The work is presented on a uniform frame work of planar consistent signed graphs.