# Interval edge colorings of some products of graphs

• Volume: 31, Issue: 2, page 357-373
• ISSN: 2083-5892

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## Abstract

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An edge coloring of a graph G with colors 1,2,...,t is called an interval t-coloring if for each i ∈ {1,2,...,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 ≥ 1 for which G has an interval t-coloring. Let ℜ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if G,H ∈ 𝔑, then the Cartesian product of these graphs belongs to 𝔑. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if G ∈ 𝔑, then G[nK₁] ∈ 𝔑 for any n ∈ ℕ. Furthermore, we show that if G,H ∈ 𝔑 and H is a regular graph, then strong and lexicographic products of graphs G,H belong to 𝔑. We also prove that tensor and strong tensor products of graphs G,H belong to 𝔑 if G ∈ 𝔑 and H is a regular graph.

## How to cite

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Petros A. Petrosyan. "Interval edge colorings of some products of graphs." Discussiones Mathematicae Graph Theory 31.2 (2011): 357-373. <http://eudml.org/doc/270969>.

@article{PetrosA2011,
abstract = {An edge coloring of a graph G with colors 1,2,...,t is called an interval t-coloring if for each i ∈ \{1,2,...,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 ≥ 1 for which G has an interval t-coloring. Let ℜ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if G,H ∈ 𝔑, then the Cartesian product of these graphs belongs to 𝔑. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if G ∈ 𝔑, then G[nK₁] ∈ 𝔑 for any n ∈ ℕ. Furthermore, we show that if G,H ∈ 𝔑 and H is a regular graph, then strong and lexicographic products of graphs G,H belong to 𝔑. We also prove that tensor and strong tensor products of graphs G,H belong to 𝔑 if G ∈ 𝔑 and H is a regular graph.},
author = {Petros A. Petrosyan},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {edge coloring; interval coloring; regular graph; products of graphs},
language = {eng},
number = {2},
pages = {357-373},
title = {Interval edge colorings of some products of graphs},
url = {http://eudml.org/doc/270969},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Petros A. Petrosyan
TI - Interval edge colorings of some products of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 357
EP - 373
AB - An edge coloring of a graph G with colors 1,2,...,t is called an interval t-coloring if for each i ∈ {1,2,...,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 ≥ 1 for which G has an interval t-coloring. Let ℜ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if G,H ∈ 𝔑, then the Cartesian product of these graphs belongs to 𝔑. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if G ∈ 𝔑, then G[nK₁] ∈ 𝔑 for any n ∈ ℕ. Furthermore, we show that if G,H ∈ 𝔑 and H is a regular graph, then strong and lexicographic products of graphs G,H belong to 𝔑. We also prove that tensor and strong tensor products of graphs G,H belong to 𝔑 if G ∈ 𝔑 and H is a regular graph.
LA - eng
KW - edge coloring; interval coloring; regular graph; products of graphs
UR - http://eudml.org/doc/270969
ER -

## References

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