A note on the interval colouring thickness of graphs

Axenovich, Maria; Girão, António; Powierski, Emil; Savery, Michael; Tamitegama, Youri

doi:10.48550/arxiv.2303.04782

Cited by 2 publications

(2 citation statements)

References 7 publications

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“…However, they are present on other two edges incident to v′. The respective vertices in V 0 are incident to two edges labeled by consecutive integers: D D (8, 7), (9,8), (10,9), (11,12), (12,11), …, ( + 19, + 20).…”

Section: ≤ ∕ ≤mentioning

confidence: 99%

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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Section: ≤ ∕ ≤mentioning

confidence: 99%

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

Axenovich

Zheng

2023

Journal of Graph Theory

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“…In [17], Asratian, Casselgren and Petrosyan suggested the following natural problem: for any positive integer k, is there a graph G with θ int (G) = k? Using a probabilistic method, Axenovich, Girão, Hollom, Portier, Powierski, Savery, Tamitegama and Versteegen [26] gave a positive answer to this problem.…”

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confidence: 99%

On Interval Coloring Thickness of Some Classes of Bipartite Graphs

Petrosyan,

Yesayan

2023

Proceedings of Computer Science and Information Technologies 2023 Conference

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A proper edge-coloring of a graph G is a mapping α : E(G) −→ N such that α(e) ̸ = α(e ′ ) for every pair of adjacent edges e and e ′ in G. An interval edge-coloring of a graph G is a proper edge-coloring satisfying that the colors on the edges incident with any vertex form an interval of integers. A graph G is interval colorable if it has an interval edge-coloring with some number of colors. It is well known that not all graphs are interval colorable; a simple example is K3. Recently, Asratian, Casselgren and Petrosyan introduced and studied a new notion, the interval coloring thickness of a graph G, denoted by θint(G), which is the minimum number of interval colorable edge-disjoint subgraphs of G the union of which is G. In particular, they proved that for any graph G on n vertices, θint(G) ≤ 2⌈ n 5 ⌉. Using a probabilistic method, Axenovich and Zheng improved this upper bound by showing that θint(G) = o(n) for any graph G on n vertices. In this paper, we study the interval coloring thickness of some classes of bipartite graphs. In particular, we determine or bound the interval coloring thickness of bipartite graphs constructed based on finite affine and projective planes.

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A note on the interval colouring thickness of graphs

Cited by 2 publications

References 7 publications

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

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

On Interval Coloring Thickness of Some Classes of Bipartite Graphs

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