Walk-Powers and Homomorphism Bounds of Planar Signed Graphs

Article Type

Research Article

Publication Title

Graphs and Combinatorics

Abstract

As an extension of the Four-Color Theorem it is conjectured by the first author that every planar graph of odd-girth at least 2 k+ 1 admits a homomorphism to the projective cube of dimension 2k, i.e., the Cayley graph PC(2k)=(Z22k,{e1,e2,… , e2k, J}) where the ei’s are the standard basis vectors of Z2d and J is the all 1 vector. Noting that PC(2 k) itself is of odd-girth 2 k+ 1 , in this work we show that if the conjecture is true, then PC(2 k) is an optimal such graph both with respect to the number of vertices and the 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 signed bipartite planar graphs of unbalanced-girth 2k. The work is presented in the uniform framework of planar consistent signed graphs.

First Page

1505

Last Page

1519

DOI

10.1007/s00373-015-1654-y

Publication Date

7-1-2016

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