Circular Backbone Colorings: on matching and tree backbones of planar graphs

Given a graph G, and a spanning subgraph H of G, a circular q-backbone k-coloring of (G,H) is a proper k-coloring c of G such that q ≤ |c(u) − c(v)| ≤ k − q, for every edge uv ∈ E(H). The circular qbackbone chromatic number of (G,H), denoted by CBCq(G,H), is the minimum integer k for which there exists a circular q-backbone k-coloring of (G,H). The Four Color Theorem implies that whenever 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 show that: 1) if G is planar and has no C4 as subgraph, and H is a linear spanning forest of G, then CBC2(G,H) ≤ 7; 2) 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; and 3) if G is planar and has no C4 nor C5 as subgraph, and H is a mathing of G, then CBC2(G,H) ≤ 5. These results partially answers questions posed by Broersma, Fujisawa and Yoshimoto (2003), and by Broersma, Fomin and Golovach (2007). It also points towards a positive answer for the Steinberg’s Conjecture.