A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on n vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most O(n/ log n) times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both fo...

A set of vertices $S$ a graph $G$ is $monophonically \ convex$ if every induced path joining two contained in $S$. The $monophonic convex hull$ $S$, $\langle S \rangle$, the smallest monophonically containing convexly independent$ $v \not\in \langle - \{v\} \rangle$ for \in S$. rank$ size largest monophonic independent $G$. We present characterization sets. Using this result, we show how to det...

1 Bipartite forbidden subgraphs We have seen the Erdős-Stone theorem which says that given a forbidden subgraph H, the extremal number of edges is ex(n,H) = 2(1−1/(χ(H)−1)+o(1))n. Here, o(1) means a term tending to zero as n → ∞. This basically resolves the question for forbidden subgraphs H of chromatic number at least 3, since then the answer is roughly cn2 for some constant c > 0. However, f...

For r≥1, the r-matching complex of a graph G, denoted Mr(G), is simplicial whose faces are subsets H⊆E(G) edge set G such that degree any vertex in induced subgraph G[H] at most r. In this article, we give closed form formula for homotopy type (n−2)-matching complete on n vertices. We also prove (n−1)-matching bipartite Kn,n equivalent to sphere dimension (n−1)2−1.

‎in this paper we give a characterization of unmixed tripartite‎ ‎graphs under certain conditions which is a generalization of a‎ ‎result of villarreal on bipartite graphs‎. ‎for bipartite graphs two‎ ‎different characterizations were given by ravindra and villarreal‎. ‎we show that these two characterizations imply each other‎.

In this thesis we study graph clustering on two well known families of graphs that arise in many applications, namely bipartite and chordal graphs. We study two speci c types of clustering problems. On the one hand we seek a partition of the vertex set of a graph into a bounded number of sets so that a prespeci ed function determined by the diameters of the induced subgraphs is minimized. On th...

To minimize the access cost in large disk arrays (RAID) Cohen et al. [5–7] introduced (d, f)-cluttered orderings of various set systems, d, f ∈ N. In case of a graph this amounts to an ordering of the edge set such that the number of points contained in any d consecutive edges is bounded by the number f . For the complete graph, Cohen et al. gave some optimal solution for small parameters d [5]...

Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information...