Petros A. Petrosyan

Interval Edge-Colorings of Cartesian Products of Graphs I

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 we first show that if G is an r-regular graph and G ∈ , then W(G⃞Pm) ≥ W(G) + W(Pm) + (m − 1)r (m ∈ N) and W(G⃞C2n) ≥ W(G) +W(C2n) + nr (n ≥ 2). Next, we investigate interval edge-colorings of grids, cylinders and tori. In particular, we prove that if G⃞H is planar and both factors have at least 3 vertices, then G⃞H N and w(G⃞H) ≤ 6. Finally, we confirm the first author’s conjecture on the n-dimensional cube Qn and show that Qn has an interval t-coloring if and only if n ≤ t ≤

Interval Non-edge-Colorable Bipartite Graphs and Multigraphs

An edge-coloring of a graph G with colors 1,...,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. In 1991, Erdi¾?s constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erdi¾?ss counterexample is the smallest in a sense of maximum degree known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

Interval edge colorings of some products of graphs

An edge co汯r楮g of a graph G w楴h co汯rs 1 ,2,...,t 楳 ca汬ed an 楮terval t-co汯r楮g 楦 for each i 2 f1,2,...,tg there 楳 at 汥ast one edge of G co汯red by i, and the co汯rs of edges 楮c楤ent to any vertex of G are d楳t楮ct and form an 楮terval of 楮tegers. A graph G 楳 楮terval co汯rab汥, 楦 there 楳 an 楮teger t � 1 for wh楣h G has an 楮terval t-co汯r楮g. Let N be the set of a汬 楮terval co汯rab汥 graphs. In 2004 Kuba汥 and G楡ro showed that 楦 G,H 2 N, then the Cartes楡n product of these graphs be汯ngs to N. A汳o, they formu污ted a s業楬ar prob汥m

Interval cyclic edge-colorings of graphs

A proper edge-coloring of a graph

A generalization of interval edge-colorings of graphs

G

On interval edge-colorings of bipartite graphs

with colors

Some results on cyclic interval edge colorings of graphs

1,\ldots,t

On resistance of graphs

is called an \emph{interval cyclic

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

t

Interval edge-colorings of composition of graphs

-coloring} if all colors are used, and the edges incident to each vertex