One-sided interval edge-colorings of bipartite graphs

Casselgren, Carl Johan; Toft, Bjarne

doi:10.1016/j.disc.2016.05.003

Cited by 7 publications

(4 citation statements)

References 21 publications

(51 reference statements)

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“…In the case of bipartite graphs with bipartition (X, Y), interval (R, t)-colorings with R ¼ X or R ¼ Y are also called one-sided interval t-colorings. Some new results on one-sided interval colorings of bipartite graphs were published in the last 5 years by Kamalian [20], Casselgren and Toft [11]. This type of coloring is of particular interest for graphs that do not have interval colorings.…”

Section: Variations and Generalizations On Interval Edge-coloringsmentioning

confidence: 99%

Computer Science – Theory and Applications

2019

Lecture Notes in Computer Science

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Section: Variations and Generalizations On Interval Edge-coloringsmentioning

confidence: 99%

Computer Science – Theory and Applications

2019

Lecture Notes in Computer Science

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“…Several sufficient conditions for a (3,4)-biregular graph to admit an interval coloring has been obtained [2,7,19,22]. In [6] we give a sufficient condition for a (3, 5)-biregular graph to admit an interval coloring.…”

Section: Introductionmentioning

confidence: 97%

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

Casselgren

Toft

2014

Journal of Graph Theory

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Abstract. A proper edge coloring of a graph with colors 1, 2, 3, . . . is called an interval coloring if the colors on the edges incident to each vertex form an interval of integers. A bipartite graph is (a, b)-biregular if every vertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. We prove that all (3, 6)-biregular graphs have interval colorings and that all (3, 9)-biregular graphs having a cubic subgraph covering all vertices of degree 9 admit interval colorings. Moreover, we prove that slightly weaker versions of the conjecture hold for (3, 5)-biregular, (4, 6)-biregular and (4, 8)-biregular graphs. All our proofs are constructive and yield polynomial time algorithms for constructing the required colorings.

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“…Several sufficient conditions for a (3,4)-biregular graph G to admit an interval 6-coloring have been obtained: Pyatkin [25] proved that if G has a 3-regular subgraph covering the vertices of degree 4, then it has an interval coloring; Yang et al [28] proved that if G is the union of two edge-disjoint (2, 3)-biregular subgraphs H 1 and H 2 such that vertices of degree 4 in G has degree 2 in H 1 and H 2 , then G has an interval coloring; G has an interval coloring if it has a spanning subgraph consisting of paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8} [4,10]. (See also [3,8] for related results.)…”

Section: Introductionmentioning

confidence: 99%

On interval and cyclic interval edge colorings of (3,5)-biregular graphs

Casselgren

Toft²

2017

Discrete Mathematics

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Dedicated to Horst Sachs on his 90th birthday. Abstract.A proper edge coloring f of a graph G with colors 1, 2, 3, . . . , t is called an interval coloring if the colors on the edges incident to every vertex of G form an interval of integers. The coloring f is cyclic interval if for every vertex v of G, the colors on the edges incident to v either form an interval or the set {1, . . . , t} \ {f (e) : e is incident to v} is an interval. A bipartite graph G is (a, b)-biregular if every vertex in one part has degree a and every vertex in the other part has degree b; it has been conjectured that all such graphs have interval colorings. We prove that every (3, 5)-biregular graph has a cyclic interval coloring and we give several sufficient conditions for a (3, 5)-biregular graph to admit an interval coloring. *

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One-sided interval edge-colorings of bipartite graphs

Cited by 7 publications

References 21 publications

Computer Science – Theory and Applications

Computer Science – Theory and Applications

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

On interval and cyclic interval edge colorings of (3,5)-biregular graphs

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