Corpus ID: 214612278

A proof of the Total Coloring Conjecture

  title={A proof of the Total Coloring Conjecture},
  author={T. Srinivasa Murthy},
  • T. Murthy
  • Published 2020
  • Mathematics, Computer Science
  • ArXiv
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
Compositions, decompositions, and conformability for total coloring on power of cycle graphs
  • Alesom Zorzi, Celina M. H. de Figueiredo, Raphael C. S. Machado, L. M. Zatesko, Uéverton S. Souza
  • Mathematics
  • 2021
Abstract Power of cycle graphs C n k have been extensively studied with respect to coloring problems, being both the vertex and the edge-coloring problems already solved in the class. The totalExpand
Hadwiger number always upper bound the chromatic number
In a simple graph G, we prove that Hadwiger number, h(G), of the given graph G always upper bound the chromatic number, χ(G), of the given graph G, that is, χ(G) ≤ h(G). This simply stated problem isExpand
Total Coloring and Total Matching: Polyhedra and Facets
This paper proposes Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and studies the strength of the corresponding Linear Programming relaxations. Expand
Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz
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 algebraicExpand
On (p, 1)-Total Labelling of NIC-Planar Graphs
  • Bei Niu, Sanyang Liu
  • Mathematics
  • 2021
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 ofExpand


Total Colourings - A survey
A survey on total coloring of graphs is given and it is shown that the total coloring can be done using at most at most $\Delta(G)+2$ colors, where $G$ is the maximum degree of $G$. Expand
Colorings of plane graphs: A survey
  • O. Borodin
  • Computer Science, Mathematics
  • Discret. Math.
  • 2013
The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors). Expand
Additive Number Theory: Inverse Problems and the Geometry of Sumsets
Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hAExpand
On colouring the nodes of a network
Let N be a network (or linear graph) such that at each node not more than n lines meet (where n > 2), and no line has both ends at the same node. Suppose also that no connected component of N is anExpand
Total Colourings of Graphs
Basic terminology and introduction.- Some basic results.- Complete r-partite graphs.- Graphs of low degree.- Graphs of high degree.- Classification of type 1 and type 2 graphs.- Total chromaticExpand
Some upper bounds on the total and list chromatic numbers of multigraphs
In this paper we discuss some estimates for upper bounds on a number of chromatic parameters of a multigraph. In particular, we show that the total chromatic number for an n-order multigraph exceedsExpand
CONTENTSIntroduction § 1. Fundamental concepts § 2. Isomorphism problems § 3. Metric questions § 4. Thickness and genus of graph § 5. Colouring problems § 6. Parts with given propertiesReferences
A Bound on the Total Chromatic Number
. The proof is probabilistic.
An improved bound for the total chromatic number of a graph, Graphs and Combinatorics, 6(1990):153-159
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Combinatorial Nullstellensatz, Combin
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