The Overfull Conjecture and the Conformability Conjecture

Abstract In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the deficiency is very small.

[1]  Abddn SANCHEZ-ARROYO,et al.  Determining the total colouring number is np-hard , 1989, Discret. Math..

[2]  Bruce A. Reed,et al.  A Bound on the Total Chromatic Number , 1998, Comb..

[3]  A. Hilton,et al.  Regular Graphs of High Degree are 1‐Factorizable , 1985 .

[4]  Anthony J. W. Hilton.  Recent results on the total chromatic number , 1993, Discret. Math..

[5]  Bruce A. Reed,et al.  On Total Colorings of Graphs , 1993, J. Comb. Theory, Ser. B.

[6]  Alexandr V. Kostochka,et al.  The total coloring of a multigraph with maximal degree 4 , 1977, Discret. Math..

[7]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[8]  Anthony J. W. Hilton,et al.  1-factorizing Regular Graphs of High Degree - an Improved Bound , 1989, Discret. Math..

[9]  Anthony J. W. Hilton.  A total-chromatic number analogue of plantholt's theorem , 1990, Discret. Math..

[10]  P. Erd Os,et al.  On the maximal number of disjoint circuits of a graph , 2022, Publicationes Mathematicae Debrecen.

[11]  Anthony J. W. Hilton,et al.  Recent progress on edge-colouring graphs , 1987, Discret. Math..

[12]  A. Hilton,et al.  Star multigraphs with three vertices of maximum degree , 1986 .

[13]  Anthony J. W. Hilton,et al.  The total chromatic number of graphs having large maximum degree , 1993, Discret. Math..

[14]  Anthony J. W. Hilton,et al.  Some refinements of the total chromatic number conjecture. , 1988 .

[15]  Bruce A. Reed,et al.  Edge coloring regular graphs of high degree , 1997, Discret. Math..

[16]  Lutz Volkmann,et al.  Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree , 1990, J. Graph Theory.

[17]  V. G. Vizing.  SOME UNSOLVED PROBLEMS IN GRAPH THEORY , 1968 .

[18]  Anthony J. W. Hilton.  Two conjectures on edge-colouring , 1989, Discret. Math..

[19]  Hung-Lin Fu,et al.  Total colorings of graphs of order 2n having maximum degree 2n−2 , 1992, Graphs Comb..