Related papers: Pairwise Suitable Family of Permutations and Boxic…

The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected…

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the…

Let $H$ be a $k$-uniform hypergraph on $n$ vertices where $n$ is a sufficiently large integer not divisible by $k$. We prove that if the minimum $(k-1)$-degree of $H$ is at least $\lfloor n/k \rfloor$, then $H$ contains a matching with…

Let $\mbox{$\cal F$}\subseteq 2^{[n]}$ be a fixed family of subsets. Let $D(\mbox{$\cal F$})$ stand for the following set of Hamming distances: $$ D(\mbox{$\cal F$}):=\{d_H(F,G):~ F, G\in \mbox{$\cal F$},\ F\neq G\}. $$ $\mbox{$\cal F$}$ is…

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We…

A graph $G$ is said to be Hamiltonian if it contains a spanning cycle. In this work, we investigate the Hamiltonian completeness of certain classes of caterpillar graphs, which are trees with a central path to which all other vertices are…

Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is…

Let $G=(V,E)$ be a connected graph. The distance between two vertices $u,v\in V$, denoted by $d(u, v)$, is the length of a shortest $u-v$ path in $G$. The distance between a vertex $v\in V$ and a subset $P\subset V$ is defined as $min\{d(v,…

Let $k$ be a positive integer and let $D$ be a digraph. A path partition $\sP$ of $D$ is a set of vertex-disjoint paths which covers $V(D)$. Its $k$-norm is defined as $\sum_{P \in \sP} \Min{|V(P)|, k}$. A path partition is $k$-optimal if…

We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph $G = (V,E)$ is $(k,p)$-planar if $V$ can be partitioned into clusters of size at most $k$ such that $G$ admits a drawing where: (i) each…

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove…

For a planar graph $H$, let $f(H)$ denote the minimum integer such that all graphs excluding $H$ as a minor have treewidth at most $f(H)$. We show that if $H$ is a disjoint union of $k$ cycles then $f(H)=O(|V(H)| + k \log k)$, which is best…

The Discrepancy of a hypergraph is the minimum attainable value, over two-colorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a…

We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a…

Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the…

We propose an approach to graph sparsification based on the idea of preserving the smallest $k$ eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend…

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlays F on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph.While it…

The boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for…

A regular partition $\mathcal{P}$ for a $3$-uniform hypergraph $H=(V,E)$ consists of a partition $V=V_1\cup \ldots \cup V_t$ and for each $ij\in {[t]\choose 2}$, a partition $K_2[V_i,V_j]=P_{ij}^1\cup \ldots \cup P_{ij}^{\ell}$, such that…

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v)…