A counterexample to the Alon-Saks-Seymour conjecture and related problems
نویسندگان
چکیده
Consider a graph obtained by taking edge disjoint union of k complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has chromatic number at most k + 1. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.
منابع مشابه
Bipartite edge partitions and the former Alon-Saks-Seymour conjecture
A famous result of Graham and Pollak states that the complete graph with n vertices can be edge partitioned into n − 1, but no fewer, complete bipartite graphs. This result has led to the study of the relationship between the chromatic and biclique partition numbers of graphs. It has become even more exciting with recent connections to the clique versus stable set problem, communication protoco...
متن کاملMore Counterexamples to the Alon-Saks-Seymour and Rank-Coloring Conjectures
The chromatic number χ(G) of a graph G is the minimum number of colors in a proper coloring of the vertices of G. The biclique partition number 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 χ(G) ≤ rank(A(G)), where rank(A(G)) is the rank of the adjacency matrix of...
متن کاملVariations on a theme of Graham and Pollak
Graham and Pollak proved that one needs at least n − 1 complete bipartite subgraphs (bicliques) to partition the edge set of the complete graph on n vertices. In this paper, we study the extension of Graham and Pollak’s result to coverings of a graph G where each edge of G is allowed to be covered a specified number of times and its generalization to complete uniform hypergraphs. We also discus...
متن کاملOn the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کاملBipartite Coverings and the Chromatic Number
Consider a graph G with chromatic number k and a collection of complete bipartite graphs, or bicliques, that cover the edges of G. We prove the following two results: • If the bipartite graphs form a partition of the edges of G, then their number is at least 2 √ log2 . This is the first improvement of the easy lower bound of log2 k, while the Alon-Saks-Seymour conjecture states that this can be...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Combinatorica
دوره 32 شماره
صفحات -
تاریخ انتشار 2010