Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of H-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical H-...

A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in way that each vertex $V_i$ adjacent to all the other $V_j$ for $1\leq i<j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote with $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring colors $1,2,\ldots,t$ an \emph{interval...

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + (x) be the outdgree and d ? (x) the indegree of the vertex x in D. The minimum (maximum) out-degree and the minimum (maximum) indegree of D are denoted by + ((+) and ? ((?), respectively. In addition, we deene = minf + ; ?...

Many distributed graph computing systems have been developed recently for efficient processing of massive graphs. These systems require many messages to be exchanged among computing machines at each step of the computation, making communication bandwidth a major performance bottleneck. We present a coded computing framework that systematically injects redundancy in the computation phase to enab...

We prove three results about the spectral radius μ (G) of a graph G : (a) Let Tr (n) be the r-partite Turán graph of order n. If G is a Kr+1-free graph of order n, then μ (G) < μ (Tr (n)) unless G = Tr (n) . (b) For most irregular graphs G of order n and size m, μ (G)− 2m/n > 1/ (2m+ 2n) . (c) Let 0 ≤ k ≤ l. If G is a graph of order n with no K2 +Kk+1 and no K2,l+1, then μ (G) ≤ min {

A digraph obtained by replacing each edge of a complete p-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicom-plete p-partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r-king is a vertex q such that every vertex in D ...

Let r ≥ 3 and (lnn)−1/(r−1) ≤ α ≤ r−3. We show that: Every r-uniform graph on n vertices with at least αnr/r! edges contains a complete r-partite graph with r − 1 parts of size ⌊ α (lnn)1/(r−1) ⌋ and one part of size ⌈ n1−α r−2 ⌉ . This result follows from a more general digraph version: Let U1, . . . , Ur be sets of size n, and M ⊂ U1 × · · · × Ur satisfy |M | ≥ αnr. If the integers s1, . . . ...

For positive integers $s,t,u,v$, we define a bipartite graph $\Gamma_{\mathbb{R}}(X^s Y^t,X^u Y^v)$ where each partite set is copy of $\mathbb{R}^3$, and vertex $(a_1,a_2,a_3)$ in the first adjacent to $[x_1,x_2,x_3]$ second if only \[ a_2 + x_2 = a_1^s x_1^t \quad \text{and} a_3+x_3=a_1^ux_1^v. \] In this paper, classify all such graphs according girth.