In this note, we prove that the alternating group A4 is a CI-group and that all disconnected Cayley graphs of A5 are CI-graphs. As a corollary, we conclude that there are exactly 22 non-isomorphic Cayley graphs of A4 • Let G be a finite group and set G# = G \ {I}. For a subset S ~ G# with S = S-l := {S-l Is E S}, the Cayley graph is the graph Cay(G, S) with vertex set G and with x and y adjacen...

Let t ≥ 26 and let F be a k-uniform hypergraph on n vertices. Suppose that |F1∩F2∩F3| ≥ t holds for all F1,F2,F3 ∈F . We prove that the size of F is at most (n−t k−t ) if p = k n satisfies p≤ 2 √ 4t +9−1 and n is sufficiently large. The above inequality for p is best possible.

Let G be an r-uniform hypergraph. The multicolor Ramsey number rk(G) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K n yields a monochromatic copy of G. Improving slightly upon results from [1], we prove that tk + 1 ≤ rk(K2,t+1) ≤ tk + k + 2, where the lower bound holds when t and k are both powers of a prime p. When t = 1, we improve the lower bo...

A hypergraph H = (V,E) is a subtree hypergraph if there is a tree T on V such that each hyperedge of E induces a subtree of T . To find a minimum size transversal for a subtree hypergraph is, in general, NP-hard. In this paper, we show that if it is possible to decide if a set of vertices W ⊆ V is a transversal in time S(n) ( where n = |V | ), then it is possible to find a minimum size transver...

In k-hypergraph matching, we are given a collection of sets of size at most k, each with an associated weight, and we seek a maximum-weight subcollection whose sets are pairwise disjoint. More generally, in k-hypergraph b-matching, instead of disjointness we require that every element appears in at most b sets of the subcollection. Our main result is a linear-programming based (k − 1 + 1 k )app...

A graph $Gamma$ is said to be vertex-transitive or edge‎- ‎transitive‎ ‎if the automorphism group of $Gamma$ acts transitively on $V(Gamma)$ or $E(Gamma)$‎, ‎respectively‎. ‎Let $Gamma=Cay(G,S)$ be a Cayley graph on $G$ relative to $S$‎. ‎Then, $Gamma$ is said to be normal edge-transitive‎, ‎if $N_{Aut(Gamma)}(G)$ acts transitively on edges‎. ‎In this paper‎, ‎the eigenvalues of normal edge-tra...

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

Tree-width of graphs is a well studied notion, which plays an important role in structural graph theory and has many algorithmic applications. Various other graph invariants are known to be the same or within a constant factor of tree-width, for example, the bramble number or tangle number of a graph [4, 5], the branch-width [5], the linkedness [4], and the number of cops required to win the ro...

A k-uniform linear path of length l, denoted by P (k) l , is a family of k-sets {F1, . . . , Fl} such that |Fi ∩ Fi+1| = 1 for each i and Fi ∩ Fj = ∅ whenever |i − j| > 1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H , denoted by exk(n,H), is the maximum number of edges in a k-uniform hypergraph F on n vertices that does not contain H as a ...

By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of vale...