A Cayley (di)graph Cay(G,S) of a group G is called normal if the right regular representation in full automorphism Cay(G,S), and CI-(di)graph for every Cay(G,T),Cay(G,S)≅Cay(G,T) implies that there σ∈Aut(G) such Sσ=T. We call an NDCI-group or NCI-group all digraphs graphs are CI-digraphs CI-graphs, respectively. prove cyclic order n only 8∤n, either = 8 8∤n.

In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and a geometric generalization to equality of Ehrhart polynomials between two convex polytopes. We then apply our results to give a new proof of Braun’s conjecture proved recently by the authors [K...

Consider a hypergraph whose vertex set is family of $n$ lines in general position the plane, and hyperedges are induced by intersections with pseudo-discs. We prove that number $t$-hyperedges bounded $O_t(n^2)$ total $O(n^3)$. Both bounds tight.

A triangle $T'$ is $\varepsilon$-similar to another $T$ if their angles pairwise differ by at most $\varepsilon$. Given a $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, B\'ar\'any F\"uredi asked determine the maximum number of triangles $h(n,T,\varepsilon)$ being in planar point set size $n$. We show that for almost all there exists $\varepsilon=\varepsilon(T)>0$ such $h(n,T,\varepsilon)=n^3/24 (1+...

Let q be a prime power. It is shown that for any hypergraph ~,~ = {F~,..., Fdtq_~)+~ } whose maximal degree is d, there exists Z ¢ ~o c ~, such that IUF~oFI =-0 (rood q). For integers d, m __ 1 let fe(m) denote the minimal t such that for any hypergraph -~ = {Fz . . . . . Ft} whose maximal degree is d, there exists ~ ¢ o~ o c Y, such that I~F~ ~oFI -= 0 (mod m). Here we determine fd(m) when m i...

Let T be a simple k-uniform hypertree with t edges. It is shown that if H is any k-uniform hypergraph with n vertices and with minimum degree at least n k−1 2k−1(k−1)! (1+o(1)), and the number of edges of H is a multiple of t then H has a T -decomposition. This result is asymptotically best possible for all simple hypertrees with at least two edges. Mathematics Subject Classification (1991): 05...

Let 1 ≤ t ≤ 7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D| ≥ t holds for all A,B,C,D ∈ F . Then we have |F | ≤ (n−t k−t ) if | k n − 2 |< ε holds for some ε > 0 and all n > n0(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”

For a subset S of a group G such that 1 / ∈ S and S = S−1, the associated Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx−1 ∈ S. Each σ ∈ Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, Sσ ). For a positive integer m, the group G is called an m-CI-group if, for all Cayley subsets S of size at most m, whenever Cay(G, S) ...

The symmetry properties of mathematical structures are often important for understanding these structures. Graphs sometimes have a large group of symmetries, especially when they have an algebraic construction such as the Cayley graphs. These graphs are constructed from abstract groups and are vertex-transitive and this is the reason for their symmetric appearance. Some Cayley graphs have even ...