Related papers: The existence of square non-integer Heffter arrays

A $(v,k;r)$ Heffter space is a resolvable $(v_r,b_k)$ configuration whose points form a half-set of an abelian group $G$ and whose blocks are all zero-sum in $G$. It was recently proved that there are infinitely many orders $v$ for which,…

Heffter arrays were introduced by Archdeacon in 2015 as an interesting link between combinatorial designs and topological graph theory. Since the initial paper on this topic, there has been a good deal of interest in Heffter arrays as well…

For a positive integer \( k \), let \( [k] = \{1, 2, \ldots, k\} \). Let \( h \) be a non-negative integer, and let \( n \) be a multiple of \( h + 1 \). Define \( H \) as the disjoint union of \( n/(h+1) \) cliques (each of size \( h + 1…

Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array in order to provide explicit constructions of $\mathbb{Z}_{v}$-regular biembeddings of complete graphs $K_v$ into orientable surfaces. In this paper, we first…

The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which starting from a single shiftable Heffter space leads to infinitely many other shiftable Heffter spaces of the…

In 2015, Archdeacon introduced the notion of Heffter arrays and showed the connection between Heffter arrays and biembedding m-cycle and an n-cycle systems on a surface. In this paper we exploit this connection and prove that for every n >=…

For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an…

An $(\alpha,\beta)$-spanner of an $n$-vertex graph $G=(V,E)$ is a subgraph $H$ of $G$ satisfying that $dist(u, v, H) \leq \alpha \cdot dist(u, v, G)+\beta$ for every pair $(u, v)\in V \times V$, where $dist(u,v,G')$ denotes the distance…

In this paper we define a new class of partially filled arrays, called $\lambda$-fold relative Heffter arrays, that are a generalisation of the Heffter arrays introduced by Archdeacon in 2015. After showing the connection of this new…

An integer linear system is a set of inequalities with integer constraints. The solution graph of an integer linear system is an undirected graph defined on the set of feasible solutions to the integer linear system. In this graph, a pair…

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form…

For integers $n\geq k\geq 1$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian and later…

Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such…

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise…

For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as…

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the…

In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In…

We consider the set $\mathcal M_n\left(\mathbb Z; H\right)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$ with…

The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In…

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$…