On Improving the Edge-Face Coloring Theorem

Abstract

In a previous paper, the authors proved a conjecture of Melnikov that the edges and faces of a plane graph of maximum degree D may be simultaneously colored with at most D 3 colors. In this paper, the theorem is reproved with a more direct technique, which also yields improvements. For D 5, the theorem is extended to multigraphs. For D ! 7, it is shown that D 2 colors suce.

Every Plane Graph of Maximum Degree 8 has an Edge-Face 9-Coloring

“…Our result is a strengthening of the ∆ = 7 result of Sanders and Zhao [7]; it can also be viewed as an extension of the recent work on ∆ = 9 [8]. The problem of finding the provably optimal upper bounds on χ ef (G) for plane graphs G with ∆(G) ∆ remains open for ∆ ∈ {4, 5, 6, 7}.…”

Planar Graphs of Maximum Degree Seven are Class I

“…If v 7 is adjacent to a 6-vertex which is adjacent to a 3-vertex y, but v 7 is not adjacent to y, then by Lemma 2.3, v 7 is adjacent to six 7-vertices, and the only charge v 7 receives is 1 from y by Rule 5. If there is a j # [4,5] such that v 7 is adjacent to a j-vertex x which is adjacent to a (9& j)-vertex y, then by Lemma 2.3, every neighbor of v 7 besides x and y is a 7-vertex, and the only charge v 7 receives is at most 1 from x and y by Rules 6 and 9, and ch$(v 7 )=0.…”

“…Combining the result of this paper, the Four Color Theorem (e.g., [2]), and a trick of Yap (see [1]), gives new proofs of two results of the authors: that every planar graph with 2=7 has a vertex-edge (total) 9-coloring [3], as well as an edge-face 9-coloring [4].…”

“…As noted by Vizing [6], if C 4 , K 4 , the octahedron, and the icosahedron have one edge subdivided each, Class II planar graphs are produced for 2 # [2, 3,4,5]. He also showed that if 2 8, then a planar graph is always Class I.…”

Edge-face coloring of plane graphs with maximum degree nine

“…, 8}. Note that Sanders and Zhao [13] have proved that plane graphs of maximum degree ≥ 7 are ( +2)-edge-face colorable. Let us briefly mention the closely related concept of total coloring: given a graph G = (V, E), we color the elements of V ∪E so that adjacent or incident elements receive different colors.…”