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Facial entire coloring of 4-minor-free graphs

Let $G$ be a plane graph. If two edges are adjacent and consecutive on the boundary walk of a face of $G$, then they are said to be facially adjacent. We call $G$ facially entire $k$-colorable if there is a mapping from $V(G)\cup E(G)\cup F(G)$ to a $k$ color set so that any two facially adjacent edges, adjacent vertices, adjacent faces, and incident elements receive different colors. The facial entire chromatic number of $G$ is defined to be the smallest integer $k$ such that $G$ is facially entire $k$-colorable. In 2016, Fabrici, Jendrol’ and Vrbjarová conjectured that every connected, loopless, bridgeless plane graph is facially entire $7$-colorable. In this paper, we give a positive answer to this conjecture for $K_4$-minor-free graphs. More specifically, we shall prove that every $K_{4}$-minor-free graph is facially entire $7$-colorable.

Combinatorial Press

Miaodi Xu Min Chen

Utilitas Mathematica

2025

Title: Facial entire coloring of 4-minor-free graphs

Description:

Let $G$ be a plane graph.

If two edges are adjacent and consecutive on the boundary walk of a face of $G$, then they are said to be facially adjacent.

We call $G$ facially entire $k$-colorable if there is a mapping from $V(G)\cup E(G)\cup F(G)$ to a $k$ color set so that any two facially adjacent edges, adjacent vertices, adjacent faces, and incident elements receive different colors.

The facial entire chromatic number of $G$ is defined to be the smallest integer $k$ such that $G$ is facially entire $k$-colorable.

In 2016, Fabrici, Jendrol’ and Vrbjarová conjectured that every connected, loopless, bridgeless plane graph is facially entire $7$-colorable.

In this paper, we give a positive answer to this conjecture for $K_4$-minor-free graphs.

More specifically, we shall prove that every $K_{4}$-minor-free graph is facially entire $7$-colorable.

.

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Facial entire coloring of 4-minor-free graphs

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