A map colour theorem for the union of graphs

Stiebitz, Michael; Škrekovski, Riste

doi:10.1016/j.jctb.2005.06.003

Cited by 5 publications

(3 citation statements)

References 21 publications

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“…The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps [13]. While working on the map coloring problem of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color a map so that no regions sharing a common border receives the same color.…”

Section: Introductionmentioning

confidence: 99%

Graph coloring: a novel heuristic based on trailing path—properties, perspective and applications in structured networks

2019

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Graph coloring is a manifestation of graph partitioning, wherein, a graph is partitioned based on the adjacency of its elements. Partitioning serves potentially as a compartmentalization for any structural problem. Vertex coloring is the heart of the problem which is to find the chromatic number of a graph. The fact that there is no general efficient solution to this problem that may work unequivocally for all graphs opens up the realistic scope for combinatorial optimization algorithms to be invoked. The algorithmic complexity of graph coloring is non-deterministic in polynomial time (NP) and hard. To the best of our knowledge, there is no algorithm as yet that procures an exact solution of the chromatic number comprehensively for any and all graphs within the polynomial (P) time domain. However, several heuristics as well as some approximation algorithms have been attempted to obtain an approximate solution for the same. Here, we present a novel heuristic, namely, the 'trailing path', which returns an approximate solution of the chromatic number within polynomial time, and, with a better accuracy than most existing algorithms. The 'trailing path' algorithm is effectively a subtle combination of the search patterns of two existing heuristics (DSATUR and Largest First), and, operates along a trailing path of consecutively connected nodes (and thereby effectively maps to the problem of finding spanning tree(s) of the graph) during the entire course of coloring, where essentially lies both the novelty and the apt of the current approach. The study also suggests that the judicious implementation of randomness is one of the keys towards rendering an improved

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Section: Introductionmentioning

confidence: 99%

Graph coloring: a novel heuristic based on trailing path—properties, perspective and applications in structured networks

2019

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“…2 and Füredi et al [5] to χ k (K n ) ≤ n + 7 k . Regarding the graphs with bounded genus, let us define χ k (S) to be the maximum of χ k (G) over all graphs G that can be embedded in the surface S. Stiebitz anď Skrekovski [14] have determined the exact values of χ 2 for all surfaces. Füredi et al [5] have shown that…”

Section: Introduction and Definitionsmentioning

confidence: 99%

k-Chromatic Number of Graphs on Surfaces

Dvořák¹,

Škrekovski²

2009

SIAM J. Discrete Math.

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A well-known result ) states that the maximum chromatic number of a graph embedded in a given surface S coincides with the size of the largest clique that can be embedded in S, and that this number can be expressed as a simple formula in the Euler genusWe derive a Heawood-type formula for the k-chromatic number of graphs embedded in a fixed surface, improving the previously known upper bounds. In infinitely many cases, the new upper bound coincides with the lower bound obtained from embedding disjoint cliques in the surface. In the proof of this result, we derive a variant of Euler's formula for the union of several graphs that might be interesting independently.

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“…Nordhaus-Gaddum-type theorems establish bounds on f (G) + f (G) for some graph parameter f , where G is the complement of a graph G. The literature has numerous examples; see [1,4,5,8,10,13,14] for a few. Our main result is the following Nordhaus-Gaddum-type theorem for treewidth 1 , which is a graph parameter of particular importance in structural and algorithmic graph theory.…”

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confidence: 99%