We describe two constructions of (very) dense graphs which are edge disjoint unions of large {\em induced} matchings. The first construction exhibits graphs on $N$ vertices with ${N \choose 2}-o(N^2)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o(1)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-δ}$ induced matchings, where $δ> 0.058$. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial and Meshulam on communicating over a shared channel, and (slightly) extends the analysis of Håstad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity. Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate rounding scheme for the directed Steiner tree problem.
We call a graph $G$ an $(r,t)$-Ruzsa-Szemerédi graph if its edge set can be partitioned into $t$ edge-disjoint induced matchings, each of size $r$. These graphs were introduced in 1978 and has been extensively studied since then. In this paper, we consider the case when $r=cn$. For $c>1/4$, we determine the maximum possible $t$ which is a constant depending only on $c$. On the other hand, when $c=1/4$, there could be as many as $Ω(\log n)$ induced matchings. We prove that this bound is tight up to a constant factor. Finally, when $c$ is fixed strictly between $1/5$ and $1/4$, we give a short p…
Let $G=(V,E)$ be a graph. If $G$ is a König graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $Δ_{G}$ is pure shellable, $R/I_Δ$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of König type $e_{1},...,e_{g}$ without square with two $e_i$'s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical…
We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemerédi graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemerédi. The graphs induced by this protocol have $n$ vertices, $Ω(n^2/\log n)$ edges, and are decomposable into $n^{1+O(1/\log \log n)}$ induced matchings. Another protocol is an explicit (and slightly simpler) version of the construction of Alon, Moitra and Sudakov, producing graphs with similar properties. We also generalize the above protocols to more than three players, in order to construct den…
Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph $G(m,n,p)=K_{m,n}-pK_2$ $(p\leq \min\{m,n\})$, which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in $G(n,n,p)$. As a corollary, we obtain the Kirchhoff index of $G(m,n,p)$ which exte…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly $n$-edge-coloured complete bipartite graphs $K_{n,n}$ which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of $K_{n,n}$ without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constra…
In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the collage such that |E \ F| < 2. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a …
