The complexity of chromatic strength and chromatic edge strength

Marx, Dániel

doi:10.1007/s00037-005-0201-2

Cited by 9 publications

(3 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Instead, we will consider the complexity class Θ p 2 , which consists of those problems solvable by a polynomialtime deterministic algorithm using NP-oracle asked for only O(log n) times. (For a detailed introduction, please, see [27].) Proposition 8.…”

Section: -Satisfiability (3-sat) Problemmentioning

confidence: 99%

On the equality of domination number and 2-domination number

Ek̇inċi

Bujtás

2024

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

The 2-domination number γ 2 (G) of a graph G is the minimum cardinality of a set D ⊆ V (G) for which every vertex outside D is adjacent to at least two vertices in D. Clearly, γ 2 (G) cannot be smaller than the domination number γ(G). We consider a large class of graphs and characterize those members which satisfy γ 2 = γ. For the general case, we prove that it is NP-hard to decide whether γ 2 = γ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.

show abstract

Section: -Satisfiability (3-sat) Problemmentioning

confidence: 99%

On the equality of domination number and 2-domination number

Ek̇inċi

Bujtás

2024

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

show abstract

“…. , n) solves the problem, hence it is in the class Θ p 2 (see [13] for a nice introduction to Θ p 2 , or the last part of [1] for short comments on its properties). However, the exact status of the problem is unknown so far.…”

Section: Conjecture 15 For Every Integermentioning

confidence: 99%

K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

Bujtás¹,

Tuza²

2016

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

A K 3 -WORM coloring of a graph G is an assignment of colors to the vertices in such a way that the vertices of each K 3 -subgraph of G get precisely two colors. We study graphs G which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161-173] who asked whether every such graph has a K 3 -WORM coloring with two colors. In fact for every integer k ≥ 3 there exists a K 3 -WORM colorable graph in which the minimum number of colors is exactly k. There also exist K 3 -WORM colorable graphs which have a K 3 -WORM coloring with two colors and also with k colors but no coloring with any of 3, . . . , k − 1 colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide k-colorability for every k ≥ 2 (and remains intractable even for graphs of maximum degree 9 if k = 3). On the other hand, we prove positive results for d-degenerate graphs with small d, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.2010 Mathematics Subject Classification. 05C15

show abstract

“…Similarly, computing the edge chromatic sum is NP-hard for bipartite graphs [9], even if the graph is also planar and has maximum degree 3 [15]. Hardness results were also given for the vertex and edge strength of a simple graph by Salavatipour [21], and Marx [16].…”

Section: Introductionmentioning

confidence: 99%

Minimum sum edge colorings of multicycles

Cardinal¹,

Ravelomanana²,

Valencia-Pabon³

2010

Discrete Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tIn the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.

show abstract

The complexity of chromatic strength and chromatic edge strength

Cited by 9 publications

References 36 publications

On the equality of domination number and 2-domination number

On the equality of domination number and 2-domination number

K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

Minimum sum edge colorings of multicycles

Contact Info

Product

Resources

About