# A proof of the Total Coloring Conjecture

@article{Murthy2020APO, title={A proof of the Total Coloring Conjecture}, author={T. Srinivasa Murthy}, journal={ArXiv}, year={2020}, volume={abs/2003.09658} }

A $total\ coloring$ of a graph G is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$ is a set of colors, satisfying the following three conditions: 1. $f(u) \neq f(v)$ for any two adjacent vertices $u, v \in V(G)$; 2. $f(e) \neq f(e')$ for any two adjacent edges $e, e' \in E(G)$; and 3. $f(v) \neq f(e)$ for any vertex $v \in V(G)$ and any edge $e \in E(G)$ that is incident to same vertex $v$. The $total\ chromatic\ number$, $\chi''(G)$, is the minimum number of colors… Expand

#### 5 Citations

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The long-standing Erdős-Faber-Lovasz conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic… Expand

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A graph is NIC-planar if it can be drawn in the plane so that there is at most one crossing per edge and two pairs of crossing edges share at most one common end vertex. A (p, 1)-total k-labelling of… Expand

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