A survey on hedetniemi's conjecture
TL;DR: The conjecture that the categorical product of two n-chromatic graphs is still nchromatic is still open, despite many dierent approaches from dierent point of views as mentioned in this paper.
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Abstract: More than 30 years ago, Hedetniemi made a conjecture which says that the categorical product of two n-chromatic graphs is still n-chromatic. The conjecture is still open, despite many dierent approaches from dierent point of views. This article surveys methods and partial results; and discuss problems related to or motivated by this conjecture.
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Citations
A new upper bound on the cyclic chromatic number
TL;DR: In this paper, the authors introduced a new parameter k*, which is the maximum number of vertices that two faces of a graph can have in common, and proved that χc ≤ max {Δ* + 3,k* + 2, Δ* + 14, 3, k* + 6, 18}.
The Equitable Coloring of Graphs
Ko-Wei Lih
- 01 Jan 1998
TL;DR: A survey of recent progress on the equitable coloring of graphs can be found in this article, where the authors pay more attention to work done on the Equitable ∆-Coloring Conjecture.
103
Equitable Coloring of Graphs
Ko-Wei Lih
- 01 Jan 2013
TL;DR: The equitable chromatic number as discussed by the authors is the smallest integer k such that a graph G is equitably k-colorable, i.e., no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one.
95
Counterexamples to Hedetniemi's conjecture
TL;DR: The chromatic number of a simple graph can be reduced to the minimum of the chromatic numbers of the graph as mentioned in this paper, where the minimum is the number of vertices in the graph.
76
Independent sets in tensor graph powers
Noga Alon,Eyal Lubetzky +1 more
TL;DR: Brown, Nowakowski, Rall, and Rall as mentioned in this paper studied the relation between the independence ratio of tensor powers and the vertex expansion ratio of independent sets of a tensor product, and showed that these two parameters are in fact equal.
References
An introduction to the category of graphs
TL;DR: The category-theoretical approach to graph theory has gained considerably in acceptance, mainly because of its most spectacular successes [41, 47, 481] as discussed by the authors, and it is the intention here to motivate further interest in this fruitful area by describing and illustrating the general approach in a reasonably self-contained and (as much as possible) nontechnical way.
56
Symmetric graphs and interpretations
Emo Welzl,Emo Welzl +1 more
TL;DR: The position of symmetric graphs in this geography of graphs is investigated and the mechanism of interpretation (subgraph homomorphism, homomorphic embedding, general coloring) is investigated.
52
Arc Colorings of Digraphs
C.C Harner,R. C. Entringer +1 more
TL;DR: In this paper, it was shown that if Tn is the transitive tournament on n points then c(Tn) = {log 2 n} but [((n + 1)2] colors suffice if the color classes are required to be oriented trees.
50
Coloring digraphs by iterated antichains
Svatopluk Poljak
- 01 Jan 1991
TL;DR: In this paper, it was shown that the minimum chromatic number of a product of two n-chromatic graphs is either bounded by 9 or tends to infinity, by coloring iterated adjoints by iterated antichains of a poset.