We introduce a special type of graph embedding called book embedding and apply it to posets. A book embedding scheme for bipartite graphs is given, and is used to extend the embeddings to general k-partite graphs. Finally, we view the Hasse diagram of a poset as a directed k-partite graph and use this scheme to derive a book embedding for arbitrary posets. Using this book embedding scheme, we a...

In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipart...

Abstract. 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...

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...

A topological book embedding of a graph is an embedding in a book that carries the vertices in the spine of the book and the edges in the pages so that edges are allowed to cross the spine. Recently, the author has shown that for an arbitrary graph G with n vertices there exists a d + 1page book embedding of G in which each edge crosses the spine logd n times. This paper improves the result for...

A sequence S is potentially Kp,1,1 graphical if it has a realization containing a Kp,1,1 as a subgraph, where Kp,1,1 is a complete 3-partite graph with partition sizes p, 1, 1. Let σ(Kp,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Kp,1,1, n) is potentially Kp,1,1 graphical. In this paper, we prove that σ(Kp,1,1, n) ≥ 2[((p + 1)(n − 1) + 2)/2] ...

Let Kl×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G1 and G2, and integer j ≥ 2, the size multipartite Ramsey number mj(G1, G2) is the smallest integer t such that every factorization of the graph Kj×t := F1 ⊕ F2 satisfies the following condition: either F1 contains G1 or F2 contains G2. In 2007, Syafrizal e...

Let A and B be two disjoint sets of points in the plane such that no three points of A ∪ B are collinear, and let n be the number of points in A. A geometric complete bipartite graph K(A, B) is a complete bipartite graph with partite sets A and B which is drawn in the plane such that each edge of K(A, B) is a straight-line segment. We prove that (i) If |B| ≥ (n + 1)(2n − 4) + 1, then the geomet...