NP completeness of finding the chromatic index of regular graphs

Leven, Daniel; Galil, Zvi

doi:10.1016/0196-6774(83)90032-9

Cited by 164 publications

(101 citation statements)

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“…Holyer [12] showed that 3-Coloring is NP-complete on line graphs. Later, Leven and Galil [18] extended this result by showing that k-Coloring is also NP-complete on line graphs for k ≥ 4. Because line graphs are claw-free, i.e., they have no induced K 1,3 , we find that for k ≥ 3, the k-Coloring problem is NP-complete for the class of H-free graphs if H is a forest that contains a vertex with degree at least 3.…”

Section: Theorem 1 ([15])mentioning

confidence: 86%

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List Coloring in the Absence of a Linear Forest

Couturier¹,

Golovach

Kratsch³

et al. 2013

Algorithmica

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The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-013-9777-0.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1, . . . , k}.Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k-Coloring can be solved in polynomial time for graphs with no induced rP1 + P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1 + P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H.

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Section: Theorem 1 ([15])mentioning

confidence: 86%

“…As explained in Section 1, the 5-Coloring problem is NP-complete for Hfree graphs whenever H is not a linear forest, due to results of Kamiński and Lozin [13] and Leven and Galil [18]. Consequently, List 5-Coloring is NPcomplete for such graph classes.…”

Section: Construction Of Gmentioning

confidence: 99%

List Coloring in the Absence of a Linear Forest

Couturier¹,

Golovach

Kratsch³

et al. 2013

Algorithmica

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“…[18]), states that every line graph is k-choosable if and only if it is k-colorable. It is known that the k-Coloring problem is NP-complete on line graphs for all k ≥ 3 [21,24]. This implies that k-Choosability is NP-complete on line graphs for every k ≥ 3, unless the List Coloring Conjecture is false.…”

Section: Introductionmentioning

confidence: 93%

Choosability on H-free graphs

Golovach

Heggernes

Hof

et al. 2013

Information Processing Letters

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details.

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“…We note that these graphs were used by Leven & Galil (1983) to reduce the edge coloring problem of multigraphs to the edge coloring of simple graphs. Vertices R s and Q t have degree k, while vertices P t have degree 1; therefore, by the following lemma, the k edges (Q t , P t ) have pairwise different colors in every k-edge-coloring of I k .…”

Section: Graphs With S (G) > χ (G)mentioning

confidence: 99%

“…has the same complexity as "Is χ (G) ≤ k?" and the latter problem is known to be NPcomplete for every k ≥ 3 (Holyer 1981;Leven & Galil 1983). However, if we want to prove that edge strength is Θ p 2 -complete (that is, harder than the chromatic index problem), then necessarily we have to consider graphs where the edge strength and the chromatic index are not the same.…”

Section: Introductionmentioning

confidence: 99%

The complexity of chromatic strength and chromatic edge strength

Marx

2006

comput. complex.

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Abstract. The sum of a coloring is the sum of the colors assigned to the vertices (assuming that the colors are positive integers). The sum Σ(G) of graph G is the smallest sum that can be achieved by a proper vertex coloring of G. The chromatic strength s(G) of G is the minimum number of colors that is required by a coloring with sum Σ(G). For every k, we determine the complexity of the question "Is s(G) ≤ k?": it is coNP-complete for k = 2 and Θ p 2 -complete for every fixed k ≥ 3. We also study the complexity of the edge coloring version of the problem, with analogous definitions for the edge sum Σ (G) and the chromatic edge strength s (G). We show that for every k ≥ 3, it is Θ p 2 -complete to decide whether s (G) ≤ k holds. As a first step of the proof, we present graphs for every r ≥ 3 with chromatic index r and edge strength r + 1. For some values of r, such graphs were not known before.

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NP completeness of finding the chromatic index of regular graphs

Cited by 164 publications

References 1 publication

List Coloring in the Absence of a Linear Forest

List Coloring in the Absence of a Linear Forest

Choosability on H-free graphs

The complexity of chromatic strength and chromatic edge strength

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