STRONG EDGE-COLORINGS OF POWERS OF PATHS AND CYCLES Текст научной статьи по специальности «Математика»

DOI 10.24412/cl-37235 -2024-1-35-37

STRONG EDGE-COLORINGS OF POWERS OF PATHS AND

CYCLES

An edge-coloring 9 of a graph G is called strong if any two edges at distance at most 1 receive different colors. The minimum number of colors required for a strong edge-coloring of a graph G is called strong chromatic index of graph G and is denoted by Xs'(G). The k-th power of the graph G has the same vertex set as G, and two distinct vertices u and v of G are adjacent in Gk if and only if their distance in G is at most k. In this paper, we determine the exact value of the strong chromatic index for the power of paths and for the power of cycles in some special case. Keywords: strong chromatic index, paths, cycles, power of graph.

Introduction

All graphs considered in this paper are finite and simple. We denote by V(G) and E(G) the sets of vertices and edges of a graph G, respectively. The degree of a vertex v e V(G) is denoted by d(v) and the maximum degree of vertices in G by

An edge-coloring of a graph G is a mapping 9: E(G)^N. 9 is called strong if any two edges at distance at most 1 receive different colors. The minimum number of colors required for a strong edge-coloring of a graph G is called strong chromatic index of graph G and is denoted by Xs'(G).

Strong edge-coloring of graphs was introduced by Fouquet and Jolivet in 1983 [1]. Later, during a seminar in Prague, Erdos and Nesetril suggested the following conjecture.

Conjecture 1. For every graph G with maximum degree A(G),

A. Drambyan1, P. Petrosyan2

Russian-Armenian (Slavonic) University 2 Yerevan State University [email protected], [email protected]

ABSTRACT

A(G).

Xs(G) <

if A(G) is even,

1

2 - 2A(G) + 1) , if A(G) is odd.

4

Conjecture was proved for A(G) = 3 by Andersen [2] and Horak [3] independently. In 2018 Huang et al. [5] showed that for A(G) = 4, Xs'(G) < 21, which is currently best-known result. Also, in 1997 Molloy and Reed [6] proved that Xs'(G)

< 1.998A(G)2 for any graph with sufficiently large maximum degree A(G), and after several improvements current best-known result is Xs'(G) < 1.772A(G)2 proved by Hurley, de Joannis de Verclos, and Kang [4].

The k-th power of the graph G has the same vertex set as G, and two distinct vertices u and v of G are adjacent in Gk if and only if their distance in G is at most k. We denote by P„ and Ck, the k-th power of paths and cycles on n vertices, respectively. Below, the figures of graphs P7 and Cg are given.

Figure 1. The graph Pf.

V, v2

Fig. 2. The graph C8¡.

Main Results

We begin our considerations with strong edge-colorings of powers of paths. Theorem 1. Let be the k-th power of the path on n vertices with n > 3k + 1.

Then

y!s(P£)=\Kk + v>.

We also consider strong edge-colorings of some powers of cycles. Theorem 2. Let be the square power of the cycle on n vertices with n > 32.

Then

x's(c£) = 9.

REFERENCES

1. Fouquet J. & Jolivet J. Strong edge-colorings of graphs and applications to multi-k-gons. Ars Combinatoria A, 16, 1983. PP. 141-150.

2. Andersen L. The strong chromatic index of a cubic graph is at most 10. Discrete Mathematics, 108 (1-3), 1992. PP. 231-252.

3. Horak P., Qing, H. & Trotter W. Induced matchings in cubic graphs // Journal of Graph Theory, 17(2), 1993. PP. 151-160.

4. Hurley E., de Joannis de Verclos, R. & Kang R. An improved procedure for colouring graphs of bounded local density. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), 2021. PP. 135-148.

5. Huang, M., Santana, M., & Yu, G. Strong chromatic index of graphs with maximum degree four. 2018. arXiv preprint arXiv:1806.07012.

6. Molloy M. & Reed B. A bound on the strong chromatic index of a graph // Journal of Combinatorial Theory, Series B, 69 (2), 1997. PP. 103-109.

СИЛЬНЫЕ РЕБЕРНЫЕ РАСКРАСКИ СТЕПЕНЕЙ ПУТЕЙ И

ЦИКЛОВ

А.К. Драмбян1, П.А. Петросян2

1 Российско-Армянский (Славянский) университет 2Ереванский государственный университет

АННОТАЦИЯ

Реберная раскраска ф графа G называется «сильной», если ребра, находящиеся на расстоянии не более 1, окрашены в различные цвета. Минимальное количество цветов, необходимое для сильной реберной раскраски графа G, называется «сильным хроматическим индексом» и обозначается через Xs'(G). k-ая степень графа G имеет то же самое множество ребер, что и граф G, и две различные вершины u и v смежны в Gk, если расстояние между этими вершинами в графе G не превосходит k. В настоящей статье найдено точное значение сильного хроматического индекса для степеней путей и для степеней циклов в некотором частном случае.

Ключевые слова: сильный хроматический индекс, пути, циклы, степень графа.