On 3-chromatic distance-regular graphs

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix. For all known distance-regular graphs it is determined whether they are 3-chromatic. A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown that such graphs, if not complete 3-partite, must have λ ≤ 1.

[1]  Norbert Sauer,et al.  The chromatic number of the product of two 4-chromatic graphs is 4 , 1985, Comb..

[2]  Willem H. Haemers,et al.  5-chromatic Strongly Regular Graphs , 2006, Discret. Math..

[3]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[4]  Dmitry Fon-Der-Flaass.  There Exists no Distance-regular Graph with Intersection Array (5, 4, 3; 1, 1, 2) , 1993, Eur. J. Comb..

[5]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[6]  Sergey V. Shpectorov,et al.  The P-geometry for M23 has no Non-trivial 2-coverings , 1990, Eur. J. Comb..

[7]  W. Haemers.  Eigenvalue techniques in design and graph theory , 1979 .

[8]  Kris Coolsaet,et al.  A Computer-Assisted Proof of the Uniqueness of the Perkel Graph , 2005, Des. Codes Cryptogr..

[9]  A. Brouwer,et al.  Near polygons and Fischer spaces , 1994 .

[10]  An De Wispelaere,et al.  Ovoids and spreads of finite classical generalized hexagons and applications , 2005 .

[11]  H. Maldeghem,et al.  Some Classes of Rank 2 Geometries , 1995 .

[12]  A. J. Hoffman,et al.  ON EIGENVALUES AND COLORINGS OF GRAPHS, II , 1970 .

[13]  Charles Payan,et al.  On the chromatic number of cube-like graphs , 1992, Discret. Math..

[14]  Ha Henny Wilbrink,et al.  The structure of near polygons with quads , 1983 .

[15]  C. Godsil.  Interesting Graphs and Their Colourings , 2003 .

[16]  Andries E. Brouwer,et al.  The uniqueness of the near hexagon on 729 points , 1982, Comb..

[17]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..