# Interval Edge-Colorings of Cartesian Products of Graphs I

@inproceedings{Khachatrian2013IntervalEO,
title={Interval Edge-Colorings of Cartesian Products of Graphs I},
author={Hrant Khachatrian and Petros A. Petrosyan and Hovhannes Tananyan},
booktitle={Discuss. Math. Graph Theory},
year={2013}
}
• Published in Discuss. Math. Graph Theory 31 January 2012
• Mathematics, Computer Science
Abstract A proper edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used and the colors of edges incident to each vertex of G form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. Let be the set of all interval colorable graphs. For a graph G ∈ , the least and the greatest values of t for which G has an interval t-coloring are denoted by w(G) and W(G), respectively. In this paper…
17 Citations

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## References

SHOWING 1-10 OF 41 REFERENCES
Interval edge-colorings of cubic graphs
This paper proves that if G is a connected cubic multigraph (a connected cubic graph) that admits an interval t-coloring, then t\leq |V(G)| +1 (t t |V (G)|), where V(G) is the set of vertices of G and the upper bounds are sharp.
A note on upper bounds for the maximum span in interval edge-colorings of graphs
• Computer Science, Mathematics
Discret. Math.
• 2012
It is shown that the upper bounds of Asratian and Kamalian's upper bound on interval t-coloring cannot be significantly improved.
Interval edge-colorings of complete graphs and n-dimensional cubes
It is shown that if n=p2^q, where p is odd, q is nonnegative, and 2n-1@?t@?4n-2-p-q, then the complete graph K"2"n has an interval t-coloring.
Investigation on Interval Edge-Colorings of Graphs
• Computer Science, Mathematics
J. Comb. Theory, Ser. B
• 1994
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
Consecutive colorings of the edges of general graphs
• Computer Science, Mathematics
Discret. Math.
• 2001
It is proved that if G has a consecutive coloring and n⩾3 then S(G)⩽2n−4 , where S( G) is the maximum number of colors allowing a consecutive dye, and the so-called deficiency of G is investigated.
Interval edge-colorings of graph products
• Computer Science, Mathematics
ArXiv
• 2011
In this paper interval edge-colorings of various graph products are investigated.
Consecutive edge-coloring of the generalized theta-graph
• Computer Science, Mathematics
Discret. Appl. Math.
• 2007
Given a graph G, an edge-coloring of G with colors 1,2,3,... is consecutive if the colors of edges incident to each vertex are distinct and form an interval of integers. The consecutive edge-coloring
Interval colorings of edges of a multigraph
• Mathematics, Computer Science
ArXiv
• 2014
The Proposition holds, since if t > w(G) then an interval on V (G) (t − 1)-coloring can be obtained from an intervalon V (g) t-coloring by recoloring with the color t − �(G), and all edges colored by t are recolored.
Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori
• Mathematics, Computer Science
ArXiv
• 2007
In this paper interval edge colorings of bipartite cylinders and bipartites tori are investigated.
Interval edge colorings of some products of graphs
• P. A. Petrosyan
• Computer Science, Mathematics
Discuss. Math. Graph Theory
• 2011
It is shown that the product of a set of graphs G,H 2 N, then the Cartes楡n product of these graphs be汯ngs to N, where N is the set of a汬 楮terval co-��rab汥 graphs.