Hrant H. Khachatrian

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.

On interval edge-colorings of bipartite graphs

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. The set of interval colorable graphs is denoted by R. Recently, Toft has conjectured that all bipartite graphs with maximum degree at most 4 are interval colorable. In this paper we prove that: 1) if G is a bipartite graph with Δ(G) ≤ 4, then G□K2 N R; 2) if G is a bipartite graph with Δ (G) = 5 and without a vertex of degree 3, then G□K2 N R; 3) if G is a bipartite graph with Δ(G) = 6 and it has a 2-factor, then G□K2 N N. In 1999, Giaro using computer-aided methods showed that all bipartite graphs on at most 14 vertices are interval colorable. On the other hand, the smallest known examples of interval non-colorable bipartite graphs have 19 vertices. In this paper we also observe that several classes of bipartite graphs of small order have an interval coloring. In particular, we show that all bipartite graphs on 15 vertices are interval colorable.

Interval edge-colorings of K1,m,n

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.

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

An interval t-coloring of a multigraph G is a proper edge coloring with colors 1, ... , t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cycl ...

Interval edge-colorings of complete graphs

An edge-coloring of a graph G with colors 1 , 2 , ź , 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 . For an interval colorable graph G , W ( G ) denotes the greatest value of t for which G has an interval t -coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of W ( K 2 n ) is known only for n ź 4 . The second author showed that if n = p 2 q , where p is odd and q is nonnegative, then W ( K 2 n ) ź 4 n - 2 - p - q . Later, he conjectured that if n ź N , then W ( K 2 n ) = 4 n - 2 - ź log 2 n ź - ź n 2 ź , where ź n 2 ź is the number of 1s in the binary representation of n .In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on W ( K 2 n ) and determine its exact values for n ź 12 .

Further results on the deficiency of graphs

Abstract A proper t -edge-coloring of a graph G is a mapping α : E ( G ) → { 1 , … , t } such that all colors are used, and α ( e ) ≠ α ( e ′ ) for every pair of adjacent edges e , e ′ ∈ E ( G ) . If α is a proper edge-coloring of a graph G and v ∈ V ( G ) , then the spectrum of a vertex v , denoted by S ( v , α ) , is the set of all colors appearing on edges incident to v . The deficiency of α at vertex v ∈ V ( G ) , denoted by d e f ( v , α ) , is the minimum number of integers which must be added to S ( v , α ) to form an interval, and the deficiency d e f ( G , α ) of a proper edge-coloring α of G is defined as the sum ∑ v ∈ V ( G ) d e f ( v , α ) . The deficiency of a graph G , denoted by d e f ( G ) , is defined as follows: d e f ( G ) = min α d e f ( G , α ) , where minimum is taken over all possible proper edge-colorings of G . For a graph G , the smallest and the largest values of t for which it has a proper t -edge-coloring α with deficiency d e f ( G , α ) = d e f ( G ) are denoted by w d e f ( G ) and W d e f ( G ) , respectively. In this paper, we obtain some bounds on w d e f ( G ) and W d e f ( G ) . In particular, we show that for any l ∈ N , there exists a graph G such that d e f ( G ) > 0 and W d e f ( G ) − w d e f ( G ) ≥ l . It is known that for the complete graph K 2 n + 1 , d e f ( K 2 n + 1 ) = n ( n ∈ N ). Recently, Borowiecka-Olszewska, Drgas-Burchardt and Haluszczak posed the following conjecture on the deficiency of near-complete graphs: if n ∈ N , then d e f ( K 2 n + 1 − e ) = n − 1 . In this paper, we confirm this conjecture.

Interval edge-colorings of K1,m,nK1,m,n

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.