Circular backbone colorings: On matching and tree backbones of planar graphs

Abstract

A (proper) k-coloring of a graph G=(V,E) is a function c:V(G)→{1,…,k} such that c(u)≠c(v) for every uv∈E(G). Given a graph G and a spanning subgraph H of G, a circular q-backbone k-coloring of (G,H) is a k-coloring c of G such that q≤|c(u)−c(v)|≤k−q for every edge uv∈E(H). The circular q-backbone chromatic number of (G,H), denoted by CBCq(G,H), is the minimum integer k for which there exists a circular q-backbonek-coloring of (G,H).The Four Color Theorem implies that if G is planar, we have CBC2(G,H)≤8. It is conjectured that this upper bound can be improved to 7 when H is a tree, and to 6 when H is a matching. In this work, we present some partial results towards these bounds.We first prove that if G is planar containing no C4 as subgraph and H is a linear spanning forest of G, then CBC2(G,H)≤7. Then, we show that if G is a plane graph having no two 3-faces sharing an edge and H is a matching of G, then CBC2(G,H)≤6. Finally, we decrease the bound and show that if G is a planar graph having no C4 nor C5 as subgraph and H is a matching of G, then CBC2(G,H)≤5. Our results partially answer some questions raised by the community. In particular, the proofs use the Discharging Method, and this fact answers questions about whether one could prove such bounds for planar graphs without using the Four Color Theorem.