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### Investigation on Interval Edge-Colorings of Graphs

A.S. Asratian, R.R. Kamalian
1994 Journal of combinatorial theory. Series B (Print)
An edge-coloring of a simple graph G with colors !  ...  . , 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)  ...  ACKNOWLEDGMENT We appreciate all the suggestions of the referees. In particular, we would like to mention that Theorem 2 has been improved by one of the referees.  ...

### Interval cyclic edge-colorings of graphs

P.A. Petrosyan, S.T. Mkhitaryan
2016 Discrete Mathematics
In this paper some properties of interval cyclic edge-colorings are investigated.  ...  A proper edge-coloring of a graph G with colors 1, , t  is called an interval cyclic t  coloring if all colors are used, and the edges incident to each vertex ( ) v V G  are colored with ( ) G d v consecutive  ...  obtained an upper bound on ( ) W G for an interval colorable graph G depending on the number of vertices of G .  ...

### On interval edge-colorings of complete tripartite graphs

Andrzej Grzesik, Hrant Khachatrian
2013 Ninth International Conference on Computer Science and Information Technologies Revised Selected Papers
An 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 are distinct and form an interval of integers  ...  A graph G is interval colorable if it has an interval t-coloring for some positive integer t.  ...  In [12] , Petrosyan investigated interval colorings of complete graphs and n-dimensional cubes.  ...

### Interval edge-colorings of graph products [article]

Petros A. Petrosyan, Hrant H. Khachatrian, Liana E. Yepremyan, Hovhannes G. Tananyan
2012 arXiv   pre-print
In this paper interval edge-colorings of various graph products are investigated.  ...  An interval t-coloring of a graph G is a proper edge-coloring of G with colors 1,2,...,t such that at least one edge of G is colored by i, i=1,2,...  ...  In this paper interval edge-colorings of various graph products are investigated.  ...

### On interval edge-colorings of outerplanar graphs [article]

Petros A. Petrosyan
2013 arXiv   pre-print
An edge-coloring of a graph \$G\$ with colors \$1,\ldots,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  ...  For an interval colorable graph \$G\$, the least value of \$t\$ for which \$G\$ has an interval \$t\$-coloring is denoted by \$w(G)\$.  ...  In [ 6] , Kamalian investigated interval colorings of complete bipartite graphs and trees.  ...

### Interval cyclic edge-colorings of graphs [article]

Petros A. Petrosyan, Sargis T. Mkhitaryan
2014 arXiv   pre-print
A proper edge-coloring of a graph G with colors 1,...  ...  We also obtain bounds on w_c(G) and W_c(G) for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.  ...  Acknowledgement We would like to thank the organizers of 7-th Cracow conference on Graph Theory "Rytro' 14"for the nice environment and working atmosphere at the conference.  ...

### Interval edge coloring of a graph with forbidden colors

Marek Kubale
1993 Discrete Mathematics
We consider herein a problem of interval coloring the edges of a graph under the restriction that certain colors cannot be used for some edges.  ...  ., Interval edge coloring of a graph with forbidden colors, Discrete Mathematics 121 (1993) 135-143. This paper is complementary to Kubale (1989) .  ...  If G is a uninomial bipartite graph and F is such that all edges with forbidden colors have one permissible interval of colors and are incident to one vertex of maximum degree, then a chromatic coloring  ...

### Interval edge-colorings of K_1,m,n [article]

Andrzej Grzesik, Hrant Khachatrian
2013 arXiv   pre-print
In this note we prove that K_1,m,n is interval edge-colorable if and only if gcd(m+1,n+1)=1. It settles in the affirmative a conjecture of Petrosyan.  ...  In [13] , Petrosyan investigated interval colorings of complete graphs and n-dimensional cubes.  ...  Recently, Petrosyan investigated interval edge-colorings of complete multipartite graphs. In particular, he proved [12] the following result: Theorem 5. If K n,...  ...

### Interval edge-colorings of complete graphs and n-dimensional cubes

P.A. Petrosyan
2010 Discrete Mathematics
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.  ...  For non-bipartite graphs, the problems of existence, construction, and bounds on the numbers of colors in interval colorings were investigated in [1] [2] [3] 5, 10, 17, 19, 21] .  ...

### Page 2566 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews
(AR-AOS-C; Erevan) Investigation on interval edge-colorings of graphs. (English summary) J. Combin. Theory Ser. B 62 (1994), no. 1, 34-43.  ...  An edge coloring of a graph G with colors 1,2,---,¢ is an interval t-coloring if each color is used at least once and for each node x the edges adjacent to x are colored with dg(x) consecutive colors (  ...

### Interval edge colorings of some products of graphs

Petros A. Petrosyan
2011 Discussiones Mathematicae Graph Theory
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  ...  A graph G is interval colorable, if there is an integer t ≥ 1 for which G has an interval t-coloring.  ...  There are many papers [17, 24, 26, 27, 28, 31, 38] devoted to edge colorings of various products of graphs. In this paper we investigate interval edge colorings of various products of graphs.  ...

### Interval colorings of complete balanced multipartite graphs [article]

Petros A. Petrosyan
2012 arXiv   pre-print
,t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers.  ...  A complete k-partite graph G is a complete balanced k-partite graph if |V_1| = |V_2| =... = |V_k|. An edge-coloring of a graph G with colors 1,...  ...  In [ 10] , Kamalian investigated interval colorings of complete bipartite graphs and trees.  ...

### Interval edge-colorings of composition of graphs [article]

Petros A. Petrosyan, Hayk H. Tepanyan
2015 arXiv   pre-print
An edge-coloring of a graph G with consecutive integers c_1,...  ...  ,c_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.  ...  Acknowledgement We would like to thank Hrant Khachatrian for his constructive suggestions on improvements of the paper.  ...

### Interval edge-colorings of cubic graphs [article]

Petros A. Petrosyan
2011 arXiv   pre-print
,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.  ...  An edge-coloring of a multigraph G with colors 1,2,...  ...  If is a connected planar graph and G G ∈ N , then 11 ( ) . 6 ( ) V G W G ≤ In this paper we investigate interval edge-colorings of cubic graphs and multigraphs.  ...

### Precoloring extension on unit interval graphs

Dániel Marx
2006 Discrete Applied Mathematics
Comput. 5 (1) (1996) 35-56], we show that the precoloring extension problem is NP-complete on unit interval graphs.  ...  In the precoloring extension problem a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper k-coloring of the graph  ...  Acknowledgements I am grateful to Katalin Friedl, Judit Csima, and the two anonymous referees for their comments that improved the presentation of the paper.  ...
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