Abstract:
For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq n -\alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $\tau(G)=n-\alpha(G)$ with high probability, that is, with probability that tends to $1$ as $n$ tends to infinity. Here we show that this conjecture is (slightly) false, proving that for most values of $n$ tending to infinity and for $G=G(n,0.5)$, $\tau(G) \leq n-\alpha(G)-1$ with high probability, and that for some sequences of values of $n$ tending to infinity $\tau(G) \leq n-\alpha(G)-2$ with probability bounded away from $0$. We also study the typical value of $\tau(G)$ for random graphs $G=G(n,p)$ with $p < 0.5$ and show that there is an absolute positive constant $c$ so that for all $p \leq c$ and for $G=G(n,p)$, $\tau(G)=n-\Theta(\alpha(G))$ with high probability.

Abstract:
For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n -\alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $bc(G)=n-\alpha(G)$ with high probability, that is, with probability that tends to 1 as $n$ tends to infinity. The first author showed that this is slightly false, proving that for most values of $n$ tending to infinity and for $G=G(n,0.5)$, $bc(G) \leq n-\alpha(G)-1$ with high probability. We prove a stronger bound: there exists an absolute constant $c>0$ so that $bc(G) \leq n-(1+c)\alpha(G)$ with high probability.

Abstract:
Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the existence of a proper colouring from the random lists for all fixed values of $k$ and growing $n$. We show that this property exhibits a sharp threshold for $k\geq 2$ and the location of the threshold is precisely $s(n)=2n$ for $k=2$, and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$.

Abstract:
A vertex $v$ of a graph $G$ is called a groupie if its degree is notless than the average of the degrees of its neighbors. In thispaper we study the influence of bipartition $(B_1,B_2)$ on groupiesin random bipartite graphs $G(B_1,B_2,p)$ with both fixed $p$ and$p$ tending to zero.

Abstract:
A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs $K_{n,n}$ into nonorientable surfaces. Such regular embedding of $K_{n,n}$ exists only when $n = 2p_1^{a_1}p_2^{a_2}... p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i \equiv \pm 1 (\mod 8)$. In this case, the number of those regular embeddings of $K_{n,n}$ up to isomorphism is $2^k$.

Abstract:
We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory.

Abstract:
We compute the Orchard crossing number, which is defined in a similar way to the rectilinear crossing number, for the complete bipartite graphs K_{n,n}.

Abstract:
A "book" with k pages consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The pagenumber of a graph G is the minimum k such that G admits a k-page embedding (that is, a k-page drawing with no edge crossings). The k-page crossing number nu_k(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the pagenumbers and k-page crossing numbers of complete bipartite graphs. We find the exact pagenumbers of several complete bipartite graphs, and use these pagenumbers to find the exact k-page crossing number of K_{k+1,n} for 3<=k<=6. We also prove the general asymptotic estimate lim_{k->oo} lim_{n->oo} nu_k(K_{k+1,n})/(2n^2/k^2)=1. Finally, we give general upper bounds for nu_k(K_{m,n}), and relate these bounds to the k-planar crossing numbers of K_{m,n} and K_n.

Abstract:
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $\mathbb{Z}_m$, $D_m$, $\mathbb{Z}_r \times \mathbb{Z}_s$ or $(\mathbb{Z}_r \times \mathbb{Z}_s) \ltimes \mathbb{Z}_2$.

Abstract:
Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in $K_{r,s}$-interval minor free bipartite graphs. We determine exact values when $r=2$ and describe the extremal graphs. For $r=3$, lower and upper bounds are given and the structure of $K_{3,s}$-interval minor free graphs is studied.