Article

# Interval edge colorings of some products of graphs

11/2009;
Source: arXiv

ABSTRACT An edge coloring of a graph $G$ with colors $1,2,\ldots ,t$ is called an interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one edge of $G$ colored by $i$, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if there is an integer $t\geq 1$ for which $G$ has an interval $t$-coloring. Let $\mathfrak{N}$ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if $G\in \mathfrak{N}$, then $G[nK_{1}]\in \mathfrak{N}$ for any $n\in \mathbf{N}$. Furthermore, we show that if $G,H\in \mathfrak{N}$ and $H$ is a regular graph, then strong and lexicographic products of graphs $G,H$ belong to $\mathfrak{N}$. We also prove that tensor and strong tensor products of graphs $G,H$ belong to $\mathfrak{N}$ if $G\in \mathfrak{N}$ and $H$ is a regular graph. Comment: 14 pages, 5 figures, minor changes

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##### Article: Investigation on Interval Edge-Colorings of Graphs
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ABSTRACT: An edge-coloring of a simple graph G with colors 1, 2,..., t is called an interval t-coloring [3] if at least one edge of G is colored by color i, i = 1, ..., t and the edges incident with each vertex x are colored by dG(x) consecutive colors, where dG(x) is the degree of the vertex x. In this paper we investigate some properties of interval colorings and their variations. It is proved, in particular, that if a simple graph G = (V, E) without triangles has an interval t-coloring, then t ≤ |V| − 1.
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ABSTRACT: An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2,4,6,8}. We provide several sufficient conditions for the existence of such a subgraph.
Journal of Graph Theory - JGT. 01/2009; 61(2):88-97.
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##### Article: Lower bounds and a tabu search algorithm for the minimum deficiency problem.
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ABSTRACT: An edge coloring of a graph G=(V,E) is a function c:E→ℕ that assigns a color c(e) to each edge e∈E such that c(e)≠c(e′) whenever e and e′ have a common endpoint. Denoting S v (G,c) the set of colors assigned to the edges incident to a vertex v∈V, and D v (G,c) the minimum number of integers which must be added to S v (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum ∑ v∈V D v (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.
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