On optimal k-fold colorings of webs and antiwebs

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1,2,...,x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by @g"k(G). We determine the exact value of this parameter when G is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which @g"k(G) attains its lower and upper bounds based on clique and integer and fractional chromatic numbers. Additionally, we extend the concept of @g-critical graphs to @g"k-critical graphs. We identify the webs and antiwebs having this property, for every integer k>=1.

[1]  Guy Kortsarz,et al.  Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees , 2002, J. Algorithms.

[2]  Leslie E. Trotter,et al.  A class of facet producing graphs for vertex packing polyhedra , 1975, Discret. Math..

[3]  S. Stahl.  n-Tuple colorings and associated graphs , 1976 .

[4]  Susanne Wetzel,et al.  Heuristics on lattice basis reduction in practice , 2002, JEAL.

[5]  Eddie Cheng,et al.  On the Facet-Inducing Antiweb-Wheel Inequalities for Stable Set Polytopes , 2002, SIAM J. Discret. Math..

[7]  Annegret Wagler,et al.  Antiwebs are rank-perfect , 2004, 4OR.

[8]  Yuri Frota,et al.  Cliques, holes and the vertex coloring polytope , 2004, Inf. Process. Lett..

[9]  Gianpaolo Oriolo,et al.  Clique-circulants and the stable set polytope of fuzzy circular interval graphs , 2008, Math. Program..

[10]  Gintaras Palubeckis.  Facet-inducing web and antiweb inequalities for the graph coloring polytope , 2010, Discret. Appl. Math..

[11]  Leslie E. Trotter,et al.  On stable set polyhedra for K1, 3-free graphs , 1981, J. Comb. Theory, Ser. B.

[12]  Fred S. Roberts,et al.  ON THE MOBILE RADIO FREQUENCY ASSIGNMENT PROBLEM AND THE TRAFFIC LIGHT PHASING PROBLEM , 1979 .

[13]  Stéphane Pérennes,et al.  On the complexity of bandwidth allocation in radio networks , 2008, Theor. Comput. Sci..

[14]  Annegret Wagler,et al.  Almost all webs are not rank-perfect , 2004, Math. Program..

[15]  Dennis P. Geller,et al.  The chromatic number and other functions of the lexicographic product , 1975 .

[16]  Sebastiano Vigna,et al.  Hardness Results and Spectral Techniques for Combinatorial Problems on Circulant Graphs , 1998 .

[17]  Pierre Hansen,et al.  Set covering and packing formulations of graph coloring: Algorithms and first polyhedral results , 2005, Discret. Optim..

[18]  Eli Gafni,et al.  Concurrency in heavily loaded neighborhood-constrained systems , 1989, ICDCS.

[19]  Oriol Serra,et al.  On the chromatic number of circulant graphs , 2009, Discret. Math..

[20]  Isabel Méndez-Díaz,et al.  A cutting plane algorithm for graph coloring , 2008, Discret. Appl. Math..

[21]  Eddie Cheng,et al.  Antiweb-wheel inequalities and their separation problems over the stable set polytopes , 2002, Math. Program..

[22]  Friedrich Eisenbrand,et al.  The stable set polytope of quasi-line graphs , 2010, Comb..

[23]  Antonio Sassano,et al.  The Rank Facets of the Stable Set Polytope for Claw-Free Graphs , 1997, J. Comb. Theory, Ser. B.

[24]  Alain Hertz,et al.  Finding the chromatic number by means of critical graphs , 2000, JEAL.

[25]  Y. Bu,et al.  k-fold coloring of planar graphs , 2010 .

[26]  E. Scheinerman,et al.  Fractional Graph Theory: A Rational Approach to the Theory of Graphs , 1997 .

[27]  Lata Narayanan,et al.  Channel assignment and graph multicoloring , 2002 .