Abstract
A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.
Highlights
A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. e total coloring conjecture (TCC) states that every simple graph G has a total (Δ(G) + 2)-coloring, where Δ(G) is the maximum degree of G. is conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6
A graph G is said to be totally k-colorable if it admits a total k-coloring. e total chromatic number of G, denoted by χt(G), is the smallest integer k such that G is totally k-colorable. e total coloring conjecture (TCC), which was proposed by Behzad [2] and Vizing [3] independently, states that every simple graph G is totally (Δ(G) + 2)-colorable, where Δ(G) is the maximum degree of G
It is known that every planar graph G with Δ(G) ≥ 7 is (Δ(G) + 2)-colorable [5]; in particular, if Δ(G) ≥ 9, χt(G) Δ(G) + 1 [6, 7]. erefore, the only open case of TCC for planar graphs is the ones with maximum degree 6
Summary
Let H be a minimal counterexample to eorem 1 in the sense that the quantity |V(H)| + |E(H)| is minimum. at is, H satisfies the following properties:. Erefore, every proper subgraph of H has total 7-coloring φ using the color set C {1, 2, . We use Cφ(v) to denote the set of colors appearing on v and its incident edges, and Cφ(v) E proof of Lemma 1 can be found in [12]. Every 4-face in H is incident with at most one 2vertex. Let f be a 3-face incident with a 2-vertex. En, every 6-vertex incident with f has only one neighbor of degree 2. Erefore, f can be extended to a total 7-coloring of H, a contradiction. 3-vertices and they are not adjacent in H (because H contains no subgraph isomorphic to a diamond), we can properly color v1 and v3 with two available colors. We obtain a 7-total-coloring of H and a contradiction
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