We study generating functions for the number of involutions of length n avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ of length k. In several interesting cases these generating functions depend only on k and can be expressed via Chebyshev polynomials of the second kind. In particular, we show that involutions of length n avoiding ...

The biclique cover number (resp. biclique partition number) of a graph G, bc(G) (resp. bp(G)), is the least number of bicliques—complete bipartite subgraphs—that are needed to cover (resp. partition) the edges of G. The local biclique cover number (resp. local biclique partition number) of a graph G, lbc(G) (resp. lbp(G)), is the least r such that there is a cover (resp. partition) of the edges...

Last lecture, we started a discussion about the independent set (“stable set”) problem. For a given graph G = (V,E), an independent set is a subset of vertices that are mutually non-adjacent. The independent set problem asks to find the maximum cardinality of such a vertex set. We saw this problem is notoriously NP-hard: it is NP-hard to approximate within a factor n1−ε for any ε > 0. However, ...

‎in this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $c_4$-free.‎

the zarankiewicz number z(b; s) is the maximum size of a subgraph of kb,b which does not contain ks,s as a subgraph. the two-color bipartite ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of kb,b with two colors contains a ks,s in the rst color or a kt,t in the second color.in this work, we design and exploit a computational method for bounding and computin...

We give a short, direct proof that given any finite group G there exist positive integers k and l and partitions α1 and α2 of {1, . . . , kl} into l subsets of size k such that (Skl)α1,α2 ∼= G. The method used will also show that given any finite group G there exists a regular bipartite graph whose automorphism group is isomorphic to G.

We revisit the notion of graph complexity introduced by Pudll ak, RR odl, and Savick y PRS]. Using their framework, we show that suuciently strong superlinear monotone lower bounds for the very special class of 2-slice functions would imply superpolynomial lower bounds for some other functions. Given an n-vertex graph G, the corresponding 2-slice function fG on n variables evaluates to zero on ...

A bipartite graph G is bipanconnected if, for any two distinct vertices x and y of G, it contains an [x, y]-path of length l for each integer l satisfying dG(x, y) ≤ l ≤ |V (G)| − 1 and 2|(l − dG(x, y)), where dG(x, y) denotes the distance between vertices x and y in G and V (G) denotes the vertex set of G. We say a bipartite graph G is bipanpositionably bipanconnected if, for any two distinct ...

Let G be a connected, undirected and simple graph. The distance Laplacian matrix L(G) is defined as L(G)=diag(Tr)−D(G), where D(G) denotes the of diag(Tr) diagonal vertex transmissions. Denote by ρL(G) spectral radius G. In this paper, we determine lower bound n-vertex bipartite graphs with diameter 4. We characterize extremal attaining bound.

Abstract Suppose G and H are bipartite graphs $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph induced between $L(v)$ $L(v')$ is matching, whenever $vv'\in E(G)$ . We show for each $\varepsilon>0$ if has maximum degree D $|L(v)| \ge (1+\varepsilon )D/\log D$ all $v\in V(G)$ , then admits an independent transversal with respect to L provided sufficiently large. This ...