Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

Abstract

A reformulation of the four color theorem is to say that K4 is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension 2k, PC(2k), (that is the Cayley graph (Z22k,{e1,e2,…,e2k,J}, where the ei's are the standard basis and J=e1+e2+⋯+e2k) is a smallest graph of odd-girth 2k+1 to which every planar graph of odd-girth at least 2k+1 admits a homomorphism. This conjecture is related to a conjecture of P. Seymour who claims that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number (more precisely, Seymour conjectured that χ′(G)=⌈χf′(G)⌉ for any planar multigraph G). Note that the restriction of Seymour's conjecture to cubic (planar) graphs is Tait's reformulation of the four color theorem.Both these conjectures are believed to be true for the larger class of K5-minor-free graphs (which includes the class of planar graphs). Motivated by these conjectures and in extension of a recent work of L. Beaudou, F. Foucaud and the second author, which studies homomorphism bounds for the class of K4-minor-free graphs, in this work we first give a necessary and sufficient condition for a graph B of odd-girth 2k+1 to admit a homomorphism from any partial t-tree of odd-girth at least 2k+1. Applying our results to the class of partial 3-trees, which is a rich subclass of K5-minor-free graphs, we prove that PC(2k) is in fact a smallest graph of odd-girth 2k+1 to which every partial 3-tree of odd-girth at least 2k+1 admits a homomorphism. We then apply this result to show that every planar (2k+1)-regular multigraph G whose dual is a partial 3-tree, and whose fractional edge-chromatic number is 2k+1, is (2k+1)-edge-colorable. Both these results are the best known supports for the general cases of the above mentioned conjectures in extension of the four color theorem.