Restraints permitting the largest number of colourings

Brown, Jason I.; Erey, Aysel

doi:10.1016/j.dam.2017.01.022

Cited by 2 publications

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“…x -coloring c of a graph G is said to be permitted by the restraint r if c(v) / ∈ r(v) for every vertex v in V (G) . The study of restrained colorings proved useful in the construction of critical graphs [17] and recently several extremal problems on this topic were studied in [4,5,11].…”

Section: Introductionmentioning

confidence: 99%

A broken cycle theorem for the restrained chromatic function

Erey¹

2019

Turk J Math

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A restraint r on a graph G is a function that assigns each vertex of the graph a finite subset of N. For each vertex v of the graph, r(v) is called the set of colors forbidden at v. A proper coloring of G is said to be permitted by a given restraint r if each vertex v of the graph receives a color that is not from its set of forbidden colors r(v). The restrained chromatic function, denoted by πr(G, x) , is a function whose evaluations at integer x values count the number of proper x-colorings of the graph G permitted by the restraint r and this function is known to be a polynomial function of x for large enough x. The restrained chromatic function πr(G, x) is a generalization of the well-known chromatic polynomial π(G, x) , as πr(G, x) = π(G, x) if r(v) = ∅ for each vertex v of the graph. Whitney's celebrated broken cycle theorem gives a combinatorial interpretation of the coefficients of the chromatic polynomial via certain subgraphs (the so-called broken cycles). We provide an extension of this result by finding combinatorial interpretations of the coefficients of the restrained chromatic function.

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Section: Introductionmentioning

confidence: 99%