Class 1 bounds for planar graphs

There are planar class 2 graphs with maximum vertex-degree k, for each k ∈ {2, 3, 4, 5}. In 1965, Vizing proved that every planar graph with ∆ ≥ 8 is class 1. He conjectured that every planar graph with ∆ ≥ 6 is a class 1 graph. This conjecture is proved for ∆ = 7, and still open for ∆ = 6. Let k ≥ 2 and G be a k-critical planar graph. The average face-degree F (G) of G is 2 |F (G)| |E(G)|. Let Σ be an embedding of G into the Euclidean plane, and v be a vertex of the boundaries of the faces f1, . . . , fn. Let d(G,Σ)(fi) be the degree of fi, F(G,Σ)(v) = 1 n(d(G,Σ)(f1) + · · ·+ d(G,Σ)(fn)) and F ((G,Σ)) = min{F(G,Σ)(v) : v ∈ V (G)}. The local average face-degree of G is F ∗(G) = max{F ((G,Σ)) : (G,Σ) is a plane graph}. Clearly, F ∗(G) ≥ 3. The paper studies the question whether there are bounds bk, b ∗ k such that if G is a k-critical graph, then F (G) ≤ bk and F ∗(G) ≤ bk. For k ≤ 5 our results give insights into the structure of planar k-critical graphs, and the results for k = 6 give support for the truth of Vizing’s planar graph conjecture for ∆ = 6.