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

Casselgren, Carl Johan; Toft, Bjarne

doi:10.1002/jgt.21841

Cited by 23 publications

(33 citation statements)

References 11 publications

(29 reference statements)

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“…Our next result requires the following lemma that was proved in [8]. For completeness, we give the short proof here as well.…”

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 95%

“…Let G be an X , Y -bigraph where each vertex in X has degree 3 and ∆(Y ) = 6. In [8] we proved that every (3, 6)-biregular graph has an interval 7-coloring. Using Lemma 3.1, this result implies the following: Proposition 3.7.…”

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 97%

“…In [8] it was proved that the problem to determine whether a (4, 8)-biregular graph has an interval 8-coloring is N P -complete. Thus in general we need at least 9 colors for an X -interval coloring of a (4, 8)-biregular graph, which means that the upper bound in Theorem 3.10 is nearly best possible.…”

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 99%

“…Note that f ′ only uses colors from the set {4, 5,7,8,10, 11} and that f ′ is interval at every vertex of X H .…”

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 99%

“…Several sufficient conditions for a (3,4)-biregular graph G to admit an interval 6-coloring have been obtained [2,17,20]; however, it is still open whether all (3,4)-biregular graphs have interval colorings. In [8] we proved that every (3,6)-biregular graph has an interval 7-coloring and in [7] it was proved that large families of (3, 5)-biregular graphs admit interval colorings.…”

Section: Introductionmentioning

confidence: 98%

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One-sided interval edge-colorings of bipartite graphs

Casselgren

Toft

2016

Discrete Mathematics

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a b s t r a c tLet G be a bipartite graph with bipartition (X, Y ). An X -interval coloring of G is a proper edge-coloring of G by integers such that the colors on the edges incident to any vertex in X form an interval. Denote by χ ′ int (G, X ) the minimum k such that G has an X -interval coloring with k colors. In this paper we give various upper and lower bounds on χ ′ int (G, X ) in terms of the vertex degrees of G. We also determine χ ′ int (G, X ) exactly for some classes of bipartite graphs G. Furthermore, we present upper bounds on χ ′ int (G, X ) for classes of bipartite graphs G with maximum degree ∆(G) at most 9: in particular, if ∆(G) = 4, then

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“…Our next result requires the following lemma that was proved in [8]. For completeness, we give the short proof here as well.…”

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 95%

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 97%

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 99%

“…Note that f ′ only uses colors from the set {4, 5,7,8,10, 11} and that f ′ is interval at every vertex of X H .…”

Section: Claim 3 the Graph F Contains Either A Vertex In Y Of Degreementioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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One-sided interval edge-colorings of bipartite graphs

Casselgren

Toft

2016

Discrete Mathematics

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Some results on cyclic interval edge colorings of graphs

Asratian

Casselgren

2017

Journal of Graph Theory

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A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree ∆(G) ≥ 4 admits a cyclic intervalWe also prove that every Eulerian bipartite graph G with maximum degree at most 8 has a cyclic interval coloring. Some results are obtained for (a, b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r −2, 2r)-biregular (r ≥ 2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

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Interval colorings of graphs—Coordinated and unstable no‐wait schedules

Axenovich

Zheng

2023

Journal of Graph Theory

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A proper edge‐coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval, that is, form a set of consecutive integers. The interval coloring thickness of a graph is the smallest number of interval colorable graphs edge‐decomposing . We prove that for any graph on vertices. This improves the previously known bound of , see Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with an interval coloring thickness strictly greater than 2, we construct bipartite graphs whose interval coloring spectrum has arbitrarily many arbitrarily large gaps. Here, an interval coloring spectrum of a graph is the set of all integers such that the graph has an interval coloring using colors. Interval colorings of bipartite graphs naturally correspond to no‐wait schedules, say for parent–teacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with participants can be coordinated in no‐wait periods. In addition, we show that for any integers and , , there is a set of pairs of parents and teachers wanting to talk to each other, such that any no‐wait schedules are unstable—they could last hours and could last hours, but there is no possible no‐wait schedule lasting hours if .

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

Cited by 23 publications

References 11 publications

One-sided interval edge-colorings of bipartite graphs

One-sided interval edge-colorings of bipartite graphs

Some results on cyclic interval edge colorings of graphs

Interval colorings of graphs—Coordinated and unstable no‐wait schedules

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