ON THE ALGORITHM OF RECOGNIZING OF INTERVAL COLORABILITY OF CACTUS GRAPHS Текст научной статьи по специальности «Математика»

DOI 10.24412/cl-37235 -2024-1 -42-43

ON THE ALGORITHM OF RECOGNIZING OF INTERVAL COLORABILITY OF CACTUS GRAPHS

L. Muradyan1, P. Petrosyan2

1Yerevan State University, 2Yerevan State University [email protected], petros [email protected]

ABSTRACT

A graph G is called a cactus if the graph is connected and every edge of the graph lies on at most one simple cycle. An edge-coloring of a graph G with integers 1,2,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph is interval colorable if it has an interval t-coloring for some positive integer t.

In this paper, we introduce an algorithm that will recognize the interval colorability of cactus graphs with complexity 0(N * (2A(a) + N2)). Keywords: interval edge-coloring, cactus, algorithms.

Introduction

All graphs considered in this paper are finite and simple. For an undirected graph G, let us denote by V(G) and E(G) the sets of vertices and edges of a graph G, respectively. Let denote by dG(v) the degree of the vertex v E V(G) and by A(G) the maximum degree of G.

An edge-coloring of a graph G is a mapping a: E(G) ^ N . a is called proper edge-coloring, if for every pair of adjacent edges e, e' E E(G), a(e) ^ a(e'). A proper edge-coloring of a graph G with integers 1,2,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G form an interval of integers.

A graph G is called a cactus if the graph is connected and every edge of the graph lies on at most one simple cycle.

In 1987 [1] Asratian and Kamalian introduced an interval edge-coloring of graphs. In [2], Kamalian investigated interval colorings of bipartite graphs and trees (trees are the edge case of cactuses). In [3], Petrosyan investigated interval edge-colorings of complete balanced multipartite graphs. In [4], authors proved that cactuses consisting of only cycles of even length are interval colorable. In [5], Axeno-vich investigated interval edge-colorings of planar graphs.

Main Results

We find an algorithm recognizing the interval colorability of cactus graphs.

Theorem. For the cactus graph G there is an algorithm recognizing whether the graph is interval colorable or not with complexity 0(N * (2Л(С) + N2)).

Let us note that for the cactus graphs with small A(G) < 10, an algorithm will work fast enough.

REFERENCES

1. Asratian A., KamalianR. Interval Colorings of Edges of a Multigraph. Appl. Math. 5 (1987), 25-34 (in Russian)

2. Kamalian R. Interval Colorings of Complete Bipartite Graphs and Trees. Preprint of the Computing Centre of the Academy of Sciences of Armenia, Yer. (1989).

3. Petrosyan P. Interval colorings of complete balanced multipartite graphs, 2012.

4. Giaro K., Kubale M., Malafiejski M. Compact scheduling in open shop with zero-one time operations, INFOR 37 (1999), 37-47.

5. AxenovichM. On interval colorings of planar graphs. Congr. Numer. 159 (2002), 77-94.

ОБ АЛГОРИТМЕ РАСПОЗНАВАНИЕ ИНТЕРВАЛЬНОЙ ОКРАШИВАЕМОСТИ КАКТУСОВ ГРАФОВ

Л. Мурадян1, П. Петросян2

Ереванский государственный университет, 2Ереванский государственный университет [email protected], [email protected]

АННОТАЦИЯ

Граф G называется кактусом, если граф связный и любое ребро принадлежит максимум одному простому циклу. Реберная раскраска графа G в цвета 1,2,...,t называется «интервальной t-раскраской», если все цвета использованы и цвета ребер, инцидентных любой вершине графа G, различны и образуют интервал целых чисел. Граф называется интервально раскрашиваемым, если он имеет интервальное t-раскраску для некоторого положительного целого числа t. В данной статье мы представляем алгоритм, который позволит определить интервальную раскрашиваемость графов кактусов с учетом сложности 0(N * (2A(G) + N2)).

Ключевые слова: интервальная реберная-раскраска, кактус, алгоритмы.