Interval Total Colorings of Bipartite Graphs

A total coloring of a graph G is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. The concept of total coloring was introduced by V. Vizing [15] and independently by M. Behzad [3]. The total chromatic number χ (G) is the smallest number of colors needed for total coloring of G. In 1965 V. Vizing and M. Behzad conjectured that χ (G) ≤ ∆(G) + 2 for every graph G [3,15], where ∆(G) is the maximum degree of a vertex in G. This conjecture became known as Total Coloring Conjecture [5]. It is known that Total Coloring Conjecture holds for cycles, for complete graphs, for bipartite graphs, for complete multipartite graphs [17], for graphs with a small maximum degree [6,11,14], for graphs with minimum degree at least 3 4 |V (G)| [5] and for planar graphs G with ∆(G) 6= 6 [5,7,13,16]. M. Rosenfeld [11] and N. Vijayaditya [14] independently proved that the total chromatic number of graphs G with ∆(G) = 3 is at most 5. A. Kostochka in [6] proved that the total chromatic number of graphs with ∆(G) ≤ 5 is at most 7. The general upper bound for the total chromatic number was obtained by M. Molloy and B. Reed [8], who proved that χ (G) ≤ ∆(G) + 10 for every graph G. The exact value of the total chromatic number is known only for paths, cycles, complete and complete bipartite graphs, n-dimensional cubes, complete multipartite graphs of odd order [5], outerplanar graphs [18] and planar graphs G with ∆(G) ≥ 9 [4,5,7,16].