A COMPUTERIZED SYSTEM FOR GRAPH THEORY, ILLUSTRATED BY PARTIAL PROOFS FOR GRAPH-COLORING PROBLEMS *

The software system KBGRAPH, which supports graphtheoretical proofs and the analysis of graph classes, is presented by developing partial proofs for two graphcoloring problems. It is shown that Reed's Conjecture, which concerns an upper bound to the chromatic number, holds for some special classes of graphs; future approaches are briefly outlined. Another strengthening of Brooks' well-known upper bound is sketched. Details about the internal derivation strategies of the program and tools offered to the users are presented, as far as needed for an understanding of the subsequent sketch of a problem-solving process. This article is written for a twofold readership: readers who want a quick overview of the knowledge-based system will find this in sections 1 – 4; for readers interested in more details of the software system, further hints on its implementation, technical data, and the availability of the program are compiled in the last section.

[1]  Nedyalko Dimov Nenov.  On the small graphs with chromatic number 5 without 4-cliques , 1998, Discret. Math..

[2]  Michael O. Albertson,et al.  On Six‐Chromatic Toroidal Graphs , 1980 .

[3]  Matej Stehlík.  Critical graphs with connected complements , 2003, J. Comb. Theory, Ser. B.

[4]  Gordon F. Royle,et al.  Small graphs with chromatic number 5: A computer search , 1995, J. Graph Theory.

[5]  Ronald D. Dutton,et al.  INGRID: A Graph Invariant Manipulator , 1989, J. Symb. Comput..

[6]  Dieter Gernert.  Experimental Results on the Efficiency of Rule-Based Systems , 1993 .

[7]  Michael Doob,et al.  Generalized line graphs , 1981, J. Graph Theory.

[8]  Carsten Thomassen.  Five-Coloring Graphs on the Torus , 1994, J. Comb. Theory, Ser. B.

[9]  Ronald J. Gould,et al.  Graph theory , 1988 .

[10]  R. L. Brooks.  On Colouring the Nodes of a Network , 1941 .

[11]  Ronald D. Dutton,et al.  A compilation of relations between graph invariants , 1985, Networks.

[12]  Ronald D. Dutton,et al.  A compilation of relations between graph invariants - supplement I , 1991, Networks.

[13]  Paul Erdös,et al.  Chromatic Number Versus Cochromatic Number in Graphs with Bounded Clique Number , 1990, Eur. J. Comb..

[14]  Bruce A. Reed,et al.  An upper bound for the chromatic number of line graphs , 2007, Eur. J. Comb..

[15]  Alexandr V. Kostochka,et al.  On an upper bound of a graph's chromatic number, depending on the graph's degree and density , 1977, J. Comb. Theory B.

[16]  Ladislav Stacho.  New upper bounds for the chromatic number of a graph , 2001 .

[17]  Landon Rabern.  On Graph Associations , 2006, SIAM J. Discret. Math..

[18]  Alexandr V. Kostochka,et al.  Excess in colour-critical graphs , 1999 .

[19]  Paul A. Catlin,et al.  Another bound on the chromatic number of a graph , 1978, Discret. Math..

[20]  Hudson V. Kronk,et al.  The chromatic number of triangle-free graphs , 1972 .

[21]  Bruce A. Reed,et al.  A Strengthening of Brooks' Theorem , 1999, J. Comb. Theory B.

[22]  Landon Rabern.  The Borodin-Kostochka Conjecture for Graphs Containing a Doubly Critical Edge , 2007, Electron. J. Comb..