Abstract
We consider planar graphs and pseudographs with the sets of vertices V, edges E, and faces F. Remind that unlike pseudographs, graphs do not contain loops and multiple edges. The weight w(e) of an edge e=(a,b) is defined to be the degree sum s(a)+s(b) of its end vertices. Kotzig proved [5] that each 3-connected planar graph contains an edge of the weight at most 13, the bound being the best possible. This result was further developed in various directions by Kotzig and his followers [1–6,8,10]. The interest to these investigations increased recently due to discovering its applications in coloring problems [1]. Thus, Kronk and Mitchem’s conjecture Xvef(G)≤Δ(G)+4 [7] on entire coloring the vertices, edges, and faces of planar graphs with the maximal degree,Δ(G) was proved [1] for Δ(G)≥12. (The case Δ(G)=3 was solved in [7].) Moreover, for Δ(G)≥18 we have the precise bound Xvef(G)≤Δ(G)+2 [1].
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