Colouring 4-cycle systems with equitably coloured blocks

AbstractA colouring of a 4-cycle system (V,B) is a surjective mapping φ:V→Γ. The elements of Γ are colours and, for each i∈Γ, the set Ci=φ−1(i) is a colour class. If |Γ|=m, we have an m-colouring of (V,B). For every B∈B, let φ(B)={φ(x)|x∈B}. We say that a block B is equitably coloured if either |φ(B)∩Ci|=0 or |φ(B)∩Ci|=2 for every i∈Γ. Let F(n) be the set of integers m such that there exists an m-coloured 4-cycle system of order n with every block equitably coloured. We prove that: •minF(n)=3 for every n≡1(mod8), n⩾17, F(9)=∅,•{m|3⩽m⩽n+3116}⊆F(n), n≡1(mod16), n⩾17,•{m|3⩽m⩽n+2316}⊆F(n), n≡9(mod16), n⩾25,•for every sufficiently large n≡1(mod8), there is an integer m̄ such that maxF(n)⩽m̄. Moreover we show that maxF(n)=m̄ for infinite values of n

Enclosings of Decompositions of Complete Multigraphs in 22-Edge-Connected rr-Factorizations

A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $\lambda K_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for some range of values for the parameters $n$, $m$, $\lambda$, $\mu$, $r$: $r=2$, $\mu>\lambda$ and either $m \geq 2n-1$, or $m=2n-2$ and $\mu = 2$ and $\lambda=1$, or $n=3$ and $m=4$. We generalize their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $\lambda K_n$ in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for every $r \geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $\lambda$ and~$\mu$.Comment: 17 pages; fixed the proof of Theorem 1.4 and other minor change

Colouring of Voloshin for ATS(v)

A mixed hypergraph is a triple H=(S,C,D), where S is the vertex set and each of C,D is a family of not-empty subsets of S, the C-edges and D-edges respectively. A strict k-colouring of H is a surjection f  from the vertex set into a set of colours {1, 2, . . . , k} so that each C-edge contains at least two distinct vertices x, y such that f(x) = f(y) and each D-edge contains at least two vertices x, y such that f(x)=f(y). For each k ∈ {1, 2, . . . , |S|}, let r_k be the number of partitions of the vertex set into k not-empty parts (the colour classes) such that the colouring constraint is satisfied on each C-edge and D-edge. The vector R(H ) = (r_1 , . . . , r_k ) is called the chromatic spectrum of H. These concepts were introduced by V. Voloshin in 1993 [6].In this paper we examine colourings of mixed hypergraphs in the case that H is an ATS(v)

Steiner systems and configurations of points

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement

Combinatorics

[no abstract available

On the colorability of bi-hypergraphs

A {\it mixed hypergraph} ${\cal H}=({\cal V},{\cal C},{\cal D})$ consists of the vertex set ${\cal V}$ and two families of subsets of $2^{{\cal V}}$: the family ${\cal C}$ of co-edges and the family ${\cal D}$ of edges. ${\cal H}$ is said to be colorable if there is a mapping $f$ from ${\cal V}$ to the set of positive integers such that $|\{f(v):v\in e\}|<|e|$ for each $e\in {\cal C}$ and $|\{f(v):v\in e\}|>1$ for each $e\in {\cal D}$. There exist mixed hypergraphs which are uncolorable, and quite little about these mixed hypergraphs is known. A mixed hypergraph is called a bi-hypergraph if its co-edge set and edge set are the same. In this article, we first apply Lov\'asz local lemma to show that any $r$-uniform bi-hypergraph with $r\ge 4$ is colorable if every edge is incident to less than $(r-1)^{r-1}e^{-1}-1$ other edges, where $e$ is the base of natural logarithms. Then, we show that among all the uncolorable $3$-uniform bi-hypergraphs, the smallest size of a minimal one is ten, which answers a question raised by Tuza and Voloshin in 2000. As an extension, we provide a minimal uncolorable $3$-uniform bi-hypergraph of order $n$ and size at most $\frac{7n}3-4$ for every $n\ge 6$.Comment: 19 pages, 4 figure