A proof of the Total Coloring Conjecture
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A total coloring of a graph G is a map f:V(G) ∪ E(G) →𝒦, where 𝒦 is a set of colors, satisfying the following three conditions: 1. f(u) ≠ f(v) for any two adjacent vertices u, v ∈ V(G); 2. f(e) ≠ f(e') for any two adjacent edges e, e' ∈ E(G); and 3. f(v) ≠ f(e) for any vertex v ∈ V(G) and any edge e ∈ E(G) that is incident to same vertex v. The total chromatic number, χ”(G), is the minimum number of colors required for a total coloring of G. Behzad and Vizing independently conjectured that, for any graph G, χ”(G)≤Δ + 2. This is one of the classic mathematical unsolved problems. In this paper, we settle this classical conjecture by proving that the totalchromatic numberχ”(G) of a graph is indeed bounded above by Δ+2. Our novel approach involves algebraic settings over finite field ℤ_p and application of combinatorial nullstellensatz.
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