An edge-coloring of a multigraph G with colors 1,2,...,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 this paper we prove 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\leq |V(G)|), where V(G) is the set of vertices of G. Moreover, if G is a connected cubic graph, G\neq K_{4}, and G has an interval t-coloring, then t\leq |V(G)| -1. We also show that these upper bounds are sharp. Finally, we prove that if G is a bipartite subcubic multigraph, then G has an interval edge-coloring with no more than four colors.
In this paper we consider graphs which are finite, undirected, and have no loops or multiple edges and multigraphs which may contain multiple edges but no loops. Let and denote the sets of vertices and edges of a graph , respectively. An biregular bipartite graph is a bipartite graph with the vertices in one part all having degree and the vertices in the other part all having degree . The degree of a vertex is denoted by , the maximum degree of a vertex in G by
. The terms and concepts that we do not define can be found in [7,10].
are denoted by and , respectively.
The concept of interval edge-coloring of multigraphs was introduced by Asratian and Kamalian [4]. In [4,5], they proved the following two theorems. m n = + - [11]. Nevertheless, for some bipartite graphs this upper bound can be improved. Recently, Asratian and Casselgren [3] proved the following
For general graphs, Kamalian proved the following Theorem 4. [12] If is a connected graph and
For regular graphs, Kamalian and Petrosyan proved the following Theorem 6. [13] If is a connected regular graph with
On the other hand, in [16], Petrosyan proved the following theorem. ( )
For planar graphs, the coefficient in upper bounds of Theorems 4-6 was improved by Axenovich. In [6], she proved the following
In this paper we investigate interval edge-colorings of cubic graphs and multigraphs. We also consider interval edgecolorings of bipartite subcubic multigraphs. (
Note that the upper bound in Theorem 11 is tight, too. The following theorem holds. Theorem 12. For any there exists a connected cubic
It is well-known that the four color theorem [1,2] is equivalent to the statement that every bridgeless cubic planar graph is 3 -edge-colorable [10]. From here and taking into account Theorem 1, we obtain the following
We also consider cubic Halin graphs. A Halin graph is a planar graph constructed from a plane embedding of a tree with at least four vertices and with no vertices of degree 2, by connecting all the leaves of the tree with a cycle, traversing the leaves in the order given by planar embedding of the tree. In particular, we proved the following result.
However, there are interval-colorable cubic graphs for which the value of the parameter is close to
For example, in [15], it was proved that for Möbius ( )
then the cubic graph 2 n M has an interval t coloring.
In [14], a similar result was proved for nprism graphs ( ) with vertices. In particular, Khchoyan [14] proved the following
,
then the cubic graph has an interval coloring.
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