Tight bounds on the chromatic sum of a connected graph

Thomassen, Carsten; Erdös, P.; Alavi, Yousef; Malde, Paresh J.; Schwenk, Allen J.

doi:10.1002/jgt.3190130310

Cited by 34 publications

(20 citation statements)

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“…Lower and upper bounds for ξ(G) are given in [18]. In their paper the authors prove three different bounds, namely: ξ(G) ≤ |V | + |E|, which holds for any graph G; √ 8|E| ≤ ξ(G) ≤ 3/2(|E| + 1) , which holds for any connected graph G; and √ 8|E| ≤ ξ(G) ≤ 3|E| , which holds for a graph G with no isolated vertices.…”

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confidence: 99%

On the Sum Coloring Problem on Interval Graphs

1999

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In this paper we study the following NP-complete problem: given an interval graph. . , V p denotes a partition of V whose subsets are ordered by nonincreasing cardinality. We present an O(mχ(G) + n log n) time ε-approximate algorithm (ε < 2) to solve the problem, where n, m, and χ(G) are the number of nodes of the interval graph, its number of cliques, and its chromatic number, respectively. The algorithm is shown to solve the problem exactly on some classes of interval graphs, namely, the proper and the containment interval graphs, and the intersection graphs of sets of "short" intervals. The problem of determining the minimum number of colors needed to achieve the minimum ξ( V 1 , V 2 , . . . , V p ) over all p-colorings of G is also addressed.

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confidence: 99%

On the Sum Coloring Problem on Interval Graphs

1999

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“…In [18], Thomassen et al showed several bounds for chromatic sum for general graphs. The first is a rather natural result of an application of a greedy algorithm:…”

Section: Results For General Graphsmentioning

confidence: 99%

“…The second bound presented in [18] is not straightforward at all and gives an upper and lower limit for the chromatic sum in terms of e, the number of edges in G. Theorem 2.6 [18]. For any connected graph with e edges,…”

Section: Results For General Graphsmentioning

confidence: 99%

The chromatic sum of a graph: history and recent developments

Kubicka

2004

International Journal of Mathematics and Mathematical Sciences

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The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum. A natural generalization of chromatic sum is optimum cost chromatic partition (OCCP) problem, where the costs of colors can be arbitrary positive numbers. Existing results about chromatic sum, strength of a graph, and OCCP problem are presented together with some recent developments. The focus is on polynomial algorithms for some families of graphs and NP-completeness issues.

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“…In the present note we provide one of those rare examples by pointing out a connection between a new concept of graph colorings, invented recently by E. Kubicka, and subgraphs obtained by edge contractions. Some basic properties of the chromatic sum have been described by Thomassen et al [4] and by Kubicka and Schwenk [3]. In the latter paper it is also observed that small chromatic number does not imply small strength.…”

Section: Introductionmentioning

confidence: 84%

Contractions and minimalk-colorability

Tuza

1990

Graphs and Combinatorics

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Abstract. Coloring the vertex set of a graph G with positive integers, the chromatic sum ~(G) of G is the minimum sum of colors in a proper coloring. The strength of G is the largest integer that occurs in every coloring whose total is ~(G). Proving a conjecture of Kubicka and Schwenk, we show that every tree of strength s has at least ((2 + v/2) =-1 -(2 -x/-2)~-l)/x/~ vertices (s >_ 2). Surprisingly, this extremal result follows from a topological property of trees. Namely, for every s > 3 there exist precisely two trees T~ and R~ such that every tree of strength at least s is edge-contractible to T~ or R,.

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Tight bounds on the chromatic sum of a connected graph

Abstract: The chromatic sum of a graph is introduced in the dissertation of Ewa Kubicka. It is the smallest possible total among all proper colorings of G using natural numbers. In this article we determine tight bounds on the chromatic sum of a connected graph with e edges.

Cited by 34 publications

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On the Sum Coloring Problem on Interval Graphs

On the Sum Coloring Problem on Interval Graphs

The chromatic sum of a graph: history and recent developments

Contractions and minimalk-colorability

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