Let H = (V, E) be a 3-uniform linear hypergraph with one hypercycle C3. We consider a blow-up hypergraph B[H]. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph B[H] of the hypergraph H, with hyperedge densities satisfying special conditions, such that the hypergraph H appears in a blow-up hypergraph as a transversal. We present an efficient...

The cyclicity of a hypergraph is an e ciently computable integer that extends the notion of the cyclomatic number of a graph The formula for the cyclicity is suggested by the join invariant of an acyclic hypergraph which is the multiset of all joining sets in any of its join trees Once we gure out how the multiplicity of a joining set depends on the structure of the acyclic hypergraph we de ne ...

A hypergraph is d-degenerate if every subhypergraph has a vertex of degree at most d. A greedy algorithm colours every such hypergraph with at most d+ 1 colours. We show that this bound is tight, by constructing an r-uniform d-degenerate hypergraph with chromatic number d + 1 for all r ≥ 2 and d ≥ 1. Moreover, the hypergraph is triangle-free, where a triangle in an r-uniform hypergraph consists...

In this paper we introduce the Cayley digraph structure. This can be considered as a generalization of Cayley digraph. We prove that all Cayley digraph structures are vertex transitive. Many graph theoretic properties are studied in terms of algebraic properties.

A famous conjecture (usually called Ryser’s conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an r-uniform r-partite hypergraph H, the inequality τ(H) 6 (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem, and in the case of r = 3, which was proved by Aharoni in 2001. Here we ...

We present a matrix multiplication based algorithm for counting the number of (induced) occurrences of a fixed r-uniform hypergraph in a larger hypergraph. In many cases, the running time is better than that of the näıve algorithm. We also present several useful applications of the algorithm, such as determining the dominant color among monochromatic simplices in a redblue edge-colored hypergra...

In this paper we establish the Cayley expansion theory on factored and shortest expansions of typical Cayley expressions in twoand three-dimensional projective geometry. We set up a group of Cayley factorization formulae based on the classification of Cayley expansions. We continue to establish three powerful simplification techniques in bracket computation. On top of the Cayley expansions and ...

Generalized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equa...

‎Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$‎. ‎Then the energy of‎ ‎$Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ‎Also‎ ‎the Estrada index of $Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$‎. ‎In this paper‎, ‎we compute the eigen...

Let G be a finite group of order n ≥ 2 and T be the Cayley table of G. Obviously T is a Latin square. In this paper we study submatrices of T which are Latin squares. Let Lm(G) where m ≤ n be the number of Latin squares of order m in T . We compute Lm(G) and study some properties of such submatrices. Also we classify the Latin squares of order m in T and compute the number of each classes in so...