More Counterexamples to the Alon-Saks-Seymour and Rank-Coloring Conjectures

The chromatic number $\chi(G)$ of a graph $G$ is the minimum number of colors in a proper coloring of the vertices of $G$. The biclique partition number ${\rm bp}(G)$ is the minimum number of complete bipartite subgraphs whose edges partition the edge-set of $G$. The Rank-Coloring Conjecture (formulated by van Nuffelen in 1976) states that $\chi(G)\leq {\rm rank}(A(G))$, where ${\rm rank}(A(G))$ is the rank of the adjacency matrix of $G$. This was disproved in 1989 by Alon and Seymour. In 1991, Alon, Saks, and Seymour conjectured that $\chi(G)\leq {\rm bp}(G)+1$ for any graph $G$. This was recently disproved by Huang and Sudakov. These conjectures are also related to interesting problems in computational complexity. In this paper, we construct new infinite families of counterexamples to both the Alon-Saks-Seymour Conjecture and the Rank-Coloring Conjecture. Our construction is a generalization of similar work by Razborov, and Huang and Sudakov.

[1]  Richard P. Stanley,et al.  Branchings and partitions , 1975 .

[2]  A. Razborov.  Communication Complexity , 2011 .

[3]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[4]  Robert Johansson,et al.  Pattern Avoidance in Alternating Sign Matrices , 2007 .

[5]  E. Kushilevitz.  Communication Complexity , 1997, Adv. Comput..

[6]  Peter Winkler,et al.  Proof of the squashed cube conjecture , 1983, Comb..

[7]  A. A. Razborov,et al.  The gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear , 1992, Discret. Math..

[8]  Noga Alon.  Decomposition of the completer-graph into completer-partiter-graphs , 1986, Graphs Comb..

[9]  Hao Huang,et al.  A counterexample to the Alon-Saks-Seymour conjecture and related problems , 2010, Comb..

[10]  Michael E. Saks,et al.  Lattices, mobius functions and communications complexity , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[11]  Vince Grolmusz.  Computing Elementary Symmetric Polynomials with a Subpolynomial Number of Multiplications , 2002, SIAM J. Comput..

[12]  Sundar Vishwanathan,et al.  A polynomial space proof of the Graham-Pollak theorem , 2008, J. Comb. Theory, Ser. A.

[13]  Siemion Fajtlowicz,et al.  On conjectures of Graffiti , 1988, Discret. Math..

[14]  Sundar Vishwanathan,et al.  A counting proof of the Graham-Pollak Theorem , 2010, Discret. Math..

[15]  David P. Robbins,et al.  Alternating Sign Matrices and Descending Plane Partitions , 1983, J. Comb. Theory, Ser. A.

[16]  M. Yannakakis.  Expressing combinatorial optimization problems by linear programs , 1991, Symposium on the Theory of Computing.

[17]  Po-Shen Loh,et al.  Algebraic Methods in Combinatorics , 2008 .

[18]  W. H. Mills,et al.  Proof of the Macdonald conjecture , 1982 .

[19]  C. V. Nuffelen,et al.  A Bound for the Chromatic Number of a Graph , 1976 .

[20]  Jacques Verstraëte,et al.  On decompositions of complete hypergraphs , 2009, J. Comb. Theory, Ser. A.

[21]  Noga Alon,et al.  A counterexample to the rank-coloring conjecture , 1989, J. Graph Theory.

[22]  D. Cvetkovic,et al.  Spectra of graphs : theory and application , 1995 .

[23]  P. Frankl,et al.  Linear Algebra Methods in Combinatorics I , 1988 .

[24]  David A. Gregory,et al.  Addressing the Petersen graph , 2002, Electron. Notes Discret. Math..

[25]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

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

[27]  J. Wrench.  Table errata: “Table of trigonometric functions (angle in grades)” (Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 357-380 (1971), 75–80), by Dušan V. Slavič , 1973 .

[28]  R. Stanley.  Theory and Application of Plane Partitions. Part 2 , 1971 .

[29]  N. S. Barnett,et al.  Private communication , 1969 .

[30]  B. Bollobás.  Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability , 1986 .

[31]  Noam Nisan,et al.  On rank vs. communication complexity , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[32]  Alfred V. Aho,et al.  On notions of information transfer in VLSI circuits , 1983, STOC.

[33]  Pavel Bleher,et al.  Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase , 2006 .

[34]  Victor W. Marek,et al.  Book review: Combinatorics, Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability by B. Bollobas (Cambridge University Press) , 1987, SGAR.

[35]  Kurt Mehlhorn,et al.  Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract) , 1982, STOC '82.

[36]  Johan Håstad.  Tensor Rank is NP-Complete , 1990, J. Algorithms.

[37]  Brian Alspach,et al.  Cycle Decompositions of Kn and Kn-I , 2001, J. Comb. Theory, Ser. B.

[38]  Aidan Roy,et al.  The chromatic number and rank of the complements of the Kasami graphs , 2007, Discret. Math..

[39]  Doron Zeilberger,et al.  Proof of the alternating sign matrix conjecture , 1994, Electron. J. Comb..

[40]  G. W. Peck,et al.  A new proof of a theorem of Graham and Pollak , 1984, Discret. Math..

[41]  J. J. Seidel,et al.  Graphs and their spectra , 1989 .

[42]  R. P. Kurshan,et al.  On the addressing problem of loop switching , 1972 .

[43]  H. Tverberg,et al.  On the decomposition of kn into complete bipartite graphs , 1982, J. Graph Theory.

[44]  J. R. Pierce,et al.  Network for block switching of data , 1972 .

[45]  Michael Doob,et al.  Spectra of graphs , 1980 .

[46]  Yoomi Rho.  A note on the Alon-Saks-Seymour coloring conjecture , 2002, Ars Comb..