The deficiency of all generalized Hertz graphs and minimal consecutively non-colourable graphs in this class

Borowiecka-Olszewska, Marta; Drgas-Burchardt, Ewa

doi:10.1016/j.disc.2015.12.028

Cited by 8 publications

(10 citation statements)

References 10 publications

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“…For Hertz's graphs, Giaro, Kubale and Malafiejski proved the following theorem. Note that this result was recently generalized by Borowiecka-Olszewska et al [8]. Using Theorems 4.5 and 4.8 we show that the following result holds.…”

Section: Constructions Using Treessupporting

confidence: 70%

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Cyclic deficiency of graphs

Asratian

Casselgren

2019

Discrete Applied Mathematics

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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. In this paper we introduce and investigate a new notion, the cyclic deficiency of a graph G, defined as the minimum number of pendant edges whose attachment to G yields a graph admitting a cyclic interval coloring; this number can be considered as a measure of closeness of G of being cyclically interval colorable. We determine or bound the cyclic deficiency of several families of graphs. In particular, we present examples of graphs of bounded maximum degree with arbitrarily large cyclic deficiency, and graphs whose cyclic deficiency approaches the number of vertices. Finally, we conjecture that the cyclic deficiency of any graph does not exceed the number of vertices, and we present several results supporting this conjecture.

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Section: Constructions Using Treessupporting

confidence: 70%

“…Recently, Petrosyan and Khachatrian [31] proved that for near-complete graphs def(K 2n+1 − e) = n − 1 (where e is an edge of K 2n+1 ), thereby confirming a conjecture of Borowiecka-Olszewska et al [7]. Further results on deficiency appear in [1,7,8,9,13,26,31].…”

Section: Introductionmentioning

confidence: 79%

Cyclic deficiency of graphs

Asratian

Casselgren

2019

Discrete Applied Mathematics

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“…Also some classes of biregular graphs (see e.g. [9,10,16,17,20]), k-trees [8], generalized Hertz graphs [6], and generalized Sevastjanov rosettes [7] have consecutive colourings. There are many papers devoted to this topic, in particular surveys which can be found in the books [1,22].…”

Section: Motivationmentioning

confidence: 99%

“…Among graphs which do not have any consecutive colouring we emphasize these ones that lack little to be consecutive colourable. Referring to this purpose the following two classes of graphs are defined and investigated in [6,7].…”

Section: Theorem 5 Every Transitive Tournament Is Consecutively Colourablementioning

confidence: 99%

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Consecutive Colouring of Oriented Graphs

et al. 2021

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We consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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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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The deficiency of all generalized Hertz graphs and minimal consecutively non-colourable graphs in this class

Cited by 8 publications

References 10 publications

Cyclic deficiency of graphs

Cyclic deficiency of graphs

Consecutive Colouring of Oriented Graphs

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

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