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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 $q\le \lvert c(u)-c(v)\rvert \le k-q$, for every edge $uv\in E(H)$. The circular $q$-backbone chromatic number of $(G,H)$, denoted by $CBC_q(G,H)$, 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 $CBC_2(G,H)\le 8$. 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 $C_4$ as subgraph, and $H$ is a linear spanning forest of $G$, then $CBC_2(G,H)\leq 7$; 2) if $G$ is a plane graph having no two 3-faces sharing an edge, and $H$ is a matching of $G$, then $CBC_2(G,H)\leq 6$; and 3) if $G$ is planar and has no $C_4$ nor $C_5$ as subgraph, and $H$ is a mathing of $G$, then $CBC_2(G,H)\leq 5$. These results partially answer 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.