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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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