Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori

and the edge set

, , , , either and or and

and the bipartite torus

If α is a proper edge coloring of the graph G then ( ) The problem of deciding whether or not a bipartite graph belongs to N was shown in [2] to be NP -complete [3,4].

It was proved in [5] that if ( )

Theorem 2 [7]. Let G be a regular graph.

In this paper interval edge colorings of bipartite cylinders and bipartite tori are investigated. The terms and concepts that we do not define can be found in [8][9][10].

( )

,

( )

Define an edge coloring α of the graph G in the following way:

, 3 2

,

Let us show that α is an interval edge ( )

First of all let us prove that for , 1, 2,

For

we define a set i F in the following way:

,

It is not hard to check that

, and, therefore for , 1, 2, , 3

Now, let us show that the edges that are incident to a vertex

where

where

, , 3 2 2,3 2 1, 3 2 ,3 2 1 where

Therefore, α is an interval edge ( )

The proof is complete.

and ( )

, 1 2 1 , .

( )

Define an edge coloring β of the graph G in the following way:

Let us show that β is an interval edge ( )

First of all let us prove that for , 1, 2,

It is not hard to check that

where