A random bipartite graph G n n p is obtained by taking two disjoint subsets of vertices A and B of cardinality n each, and by connecting each pair of vertices a ! A and b ! B by an edge randomly and independently with probability p " p n . We show that the choice number of G n n p is, almost surely, 1 # o 1 log2 np for all values of the edge probability p " p n , where the o 1 term tends to 0 a...

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...

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R2. First, we propose an O(n2 log n) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangle...

Bipartite graphs G = (L, R; E) and H = (L, R; E) are bi-placeabe if there is a bijection f : L ∪ R → L ∪ R such that f(L) = L and f(u)f(v) / ∈ E for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H | = 2p ≥ 4 such that the sizes of G and H satisfy ‖ G ‖≤ 2p− 3 and ‖ H ‖≤ 2p− 2, and the maximum degree of H is at most 2, then G and H are bi-placeable...

An atom of a regular language L with n (left) quotients is a non-empty intersection of uncomplemented or complemented quotients of L, where each of the n quotients appears in a term of the intersection. The quotient complexity of L, which is the same as the state complexity of L, is the number of quotients of L. We prove that, for any language L with quotient complexity n, the quotient complexi...

Let G $G$ be a finite group and recall that the Frattini subgroup Frat ( ) ${\rm Frat}(G)$ is intersection of all maximal subgroups . In this paper, we investigate number , denoted α $\alpha (G)$ which minimal whose coincides with earlier work, studied in special case where simple here extend analysis to almost groups. particular, prove ⩽ 4 (G) \leqslant 4$ for every best possible. We also esta...

(a) Perhaps the most important topic in extremal graph theory, is to determine the Turán number of a graph H . This is the maximal number of edges a graph of size n can have, without having a subgraph isomorphic to H . When H is non-bipartite, the asymptotics of this function is determined by the celebrated Erdös-Stone theorem, up to an 1 + o(1) factor. However, it is a longstanding problem to ...

If m(n,l) denotes the maximum number of subsets of an n-element set such that the intersection of any two of them has cardinality divisible by l, then a trivial construction shows that m(n,l) ≥ 2 [n / l]. For l = 2, this was known to be essentially best possible. For l ≥ 3, we show by construction that m(n,l) 2 − [n / l] grows exponentially in n, and we provide upper bounds.

Let I and O denote two sets of vertices, where I ∩ O = Φ, |I| = n, |O| = r, and Bu(n, r) denote the set of unlabeled graphs whose edges connect vertices in I and O. It is shown that the following two-sided equality holds. ( r+2n−1 r ) n! ≤ |Bu(n, r)| ≤ 2 ( r+2n−1 r )

We first show that for any bipartite graph H with at most five vertices, there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-...