SOME BOUNDS ON THE NUMBER OF COLORS IN INTERVAL EDGE-COLORINGS OF GRAPHS

Authors

DOI:

https://doi.org/10.46991/PYSU:A.2024.58.2.057

Keywords:

edge-coloring, interval edge-coloring, k-connected graph, bipartite graph, dominating vertex

Abstract

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval \lb $t$-coloring}, if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A vertex $v$ of a graph $G=(V,E)$ is called a dominating vertex if $d_{G}(v)=|V|-1$, where $d_{G}(v)$ is the degree of $v$ in $G$. In this paper we prove, that if $G$ is a graph with the dominating vertex $u$ and it has an interval $t$-coloring, then $t\leq |V|+2\Delta(G-u)-1$, where $\Delta(G)$ is the maximum degree of $G$. We also show, that if a $k$-connected graph $G=(V,E)$ admits an interval $t$-coloring, then $t\leq 1+\left(\left\lfloor \dfrac{|V|-2}{k}\right\rfloor+2\right)(\Delta(G)-1)$. Moreover, if $G$ is also bipartite, then this upper bound can be improved to $t\leq 1+\left(\left\lfloor \dfrac{|V|-2}{k}\right\rfloor+1\right)(\Delta(G)-1)$. Finally, we discuss the sharpness of the obtained upper bounds on the number of colors in interval edge-colorings of these graphs.

References

West D.B. Introduction to Graph Theory. New Jersey, Prentice-Hall (2001). https://dwest.web.illinois.edu/igt/

Asratian A.S., Kamalian R.R. Interval Colorings of Edges of a Multigraph. Appl. Math. 5 (1987), 25-34 (in Russian). https://arxiv.org/abs/1401.8079

Asratian A.S., Kamalian R.R. Investigation on Interval Edge-Colorings of Graphs. J. Combin. Theory Ser. B 62 (1994), 34-43.

https://doi.org/10.1006/jctb.1994.1053

Kamalian R.R. Interval Colorings of Complete Bipartite Graphs and Trees. Yerevan, Comp. Cen. of Acad. Sci. of Armenian SSR (1989) (in Russian). https://arxiv.org/pdf/1308.2541

Petrosyan P.A. Interval Edge-Colorings of Complete Graphs and n-Dimensional Cubes. Discrete Math. 310 (2010), 1580-1587. https://doi.org/10.1016/j.disc.2010.02.001

Petrosyan P.A., Khachatrian H.H., Tananyan H.G. Interval Edge-Colorings of Cartesian Products of Graphs I. Discuss. Math. Graph Theory 33 (2013), 613-632. https://doi.org/10.7151/dmgt.1693

Giaro K. The Complexity of Consecutive Δ-Coloring of Bipartite Graphs: 4 is Easy, 5 is Hard. Ars Combinatoria 47 (1997), 287-298. https://combinatorialpress.com/ars/vol47/

Sevastjanov S.V. Interval Colorability of the Edges of a Bipartite Graph. Metodi Diskret. Analiza 50 (1990), 61-72 (in Russian).

Petrosyan P.A., Khachatrian H.H. Interval Non-Edge-Colorable Bipartite Graphs and Multigraphs. J. Graph Theory 76 (2014), 200-216. https://doi.org/10.1002/jgt.21759

Giaro K., Kubale M., Malafiejski M. Consecutive Colorings of the Edges of General Graphs. Discrete Math. 236 (2001), 131-143. https://doi.org/10.1016/S0012-365X(00)00437-4

Kamalian R.R. Interval Edge Colorings of Graphs. Doctoral Thesis. Novosibirsk (1990).

Axenovich M.A. On Interval Colorings of Planar Graphs. Congr. Numer. 159 (2002), 77-94. https://www.math.kit.edu/iag6/~axenovich/media/interval_coloring.pdf

Axenovich M, Girão A., et al. A Note on Interval Colourings of Graphs. European Journal of Combinatorics 120 (2024). Article Number 103956. https://doi.org/10.1016/j.ejc.2024.103956

Hambardzumyan A., Muradyan L. Upper Bounds on the Number of Colors in Interval Edge-Colorings of Graphs.

Discrete Math. 348 (2025). Article Number 114229. https://doi.org/10.1016/j.disc.2024.114229

Kamalian R.R., Petrosyan P.A. A Note on Upper Bounds for the Maximum Span in Interval Edge-Colorings of Graphs. Discrete Math. 312 (2012), 1393-1399. https://doi.org/10.1016/j.disc.2012.01.005

Casselgren C.J., Khachatrian H.H., Petrosyan P.A. Some Bounds on the Number of Colors in Interval and Cyclic Interval Edge Colorings of Graphs. Discrete Math. 341 (2018), 627-637. https://doi.org/10.1016/j.disc.2017.11.001

Muradyan L. On Interval Edge-Colorings of Complete Multipartite Graphs. Proc. of the YSU. Phys. and Math. Sci. 56 (2022), 19-26. https://doi.org/10.46991/PYSU:A/2022.56.1.019

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Published

2024-10-30

How to Cite

Petrosyan, P. A., & Muradyan, L. N. (2024). SOME BOUNDS ON THE NUMBER OF COLORS IN INTERVAL EDGE-COLORINGS OF GRAPHS. Proceedings of the YSU A: Physical and Mathematical Sciences, 58(2 (264), 57–65. https://doi.org/10.46991/PYSU:A.2024.58.2.057