The closed neighborhood $N[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having a common end-vertex with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1,1\}$. If $\sum _{e\in N[e]}f(x)\ge 1$ for each $e\in E(G)$, then $f$ is called a signed edge dominating function of $G$. The minimum of the values $\sum _{e\in E(G)}f(e)$, taken over all signed edge dominating functions $f$ of $G$, is called the signed edge domination number of $G$ and is denoted by ${\gamma }_s^{\prime }(G)$. It has been conjectured that ${\gamma }_s^{\prime }(T)\ge 1$ for every tree $T$. In this paper we prove that this conjecture is true and then classify all trees $T$ with ${\gamma }_s^{\prime }(T)=1,2$ and $3$.

This article is a contribution to the study of block-transitive automorphism groups of $2$-$(v,k,1)$ block designs. Let $\mathcal{D}$ be a $2$-$(v,k,1)$ design admitting a block-transitive, point-primitive but not flag-transitive group $G$ of automorphisms. Let $k_1=(k,v-1)$ and $q=p^f$ for prime $p$. In this paper we prove that if $G$ and $D$ are as above and $q>(2(k_{r}k-k_r+1)f{)}^{\frac{1}{4}}$ then $G$ does not admit a Chevalley group $E_7(q)$ as its socle.

A graph $G$ is called super edge-magic if there exists a bijection $f$ from $V(G)\cup E(G)$ to $\{1,2,\dots ,|V(G)|+|E(G)|\}$ such that $f(u)+f(v)+f(uv)=C$ is a constant for any $uv\in E(G)$ and $f(V(G))=\{1,2,\dots ,|V(G)|\}$, $f(E(G))=\{|V(G)|+1,|V(G)|+2,\dots ,|V(G)|+|E(G)|\}$. R. M. Figueroa-Centeno et al. provided the following conjecture: For every integer $n\ge 5$, the book $B_n$ is super edge-magic if and only if $n$ is even or $n\equiv 5(\mathrm{mod}8)$. In this paper, we show that $B_n$ is super edge-magic for even $n\ge 6$.

It was conjectured in $[10]$ that the upper bound for the strong chromatic index $s^{\prime }(G)$ of bipartite graphs is $\mathrm{\Delta }(G{)}^2+1$, where $\mathrm{\Delta }(G)$ is the largest degree of vertices in $G$. In this note we study the strong edge coloring of some classes of bipartite graphs that belong to the class of partial cubes. We introduce the concept of $\mathrm{\Theta }$-graph $\mathrm{\Theta }(G)$ of a partial cube $G$, and show that $s^{\prime }(G)\le \chi (\mathrm{\Theta }(G))$ for every tree-like partial cube $G$. As an application of this bound we derive that $s^{\prime }(G)\le 2\mathrm{\Delta }(G)$ if $G$ is a $p$-expansion graph.

We introduce notions of $k$-chromatic uniqueness and $k$-chromatic equivalence in the class of all Sperner hypergraphs. They generalize the chromatic uniqueness and equivalence defined in the class of all graphs $[10]$ and hypergraphs $[2,4,8]$. Using some known facts, concerning a $k$-chromatic polynomial of a hypergraph $[5]$, a set of hypergraphs whose elements are $3$-chromatically unique is indicated. A set of hypergraphs characterized by a described $3$-chromatic polynomial is also shown. The application of the investigated notions can be found in $[5]$.

A graph-pair of order $t$ is two non-isomorphic graphs $G$ and $H$ on $t$ non-isolated vertices for which $G\cup H\cong K_t$ for some integer $t\ge 4$. Given a graph-pair $(G,H)$, we say $(G,H)$ divides some graph $K$ if the edges of $K$ can be partitioned into copies of $G$ and $H$ with at least one copy of $G$ and at least one copy of $H$. We will refer to this partition as a $(G,H)$-multidecomposition of $K$.

Let $V$ denote the $n$-dimensional row vector space over a finite field ${\mathbb{F}}_q$, and let $W$ be a subspace of dimension $n-d$. Let $L(n,d)=\mathcal{P}\cup \{0\}$, where $P=\{A|A\text{ is a subspace of }V,A+W=V\}$. Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.

A graph $G$ is called $H$-equipackable if every maximal $H$-packing in $G$ is also a maximum $H$-packing in $G$. All $M_2$-equipackable graphs and $P_3$-equipackable graphs have been characterized. In this paper, $P_k$-equipackable paths, $P_k$-equipackable cycles, $M_3$-equipackable paths and $M_3$-equipackable cycles are characterized.

Let $G$ be a graph with $r$ vertices of degree at least two. Let $H$ be any graph. Consider $r$ copies of $H$. Then $G\oplus H$ denotes the graph obtained by merging the chosen vertex of each copy of $H$ with every vertex of degree at least two of $G$. Let $T_0$ and $T^{A_1}$ be any two caterpillars. Define the first attachment tree $T_1=T_0\oplus T^{A_1}$. For $i\ge 2$, define recursively the $(i^{th})$ attachment tree $T_i=T_{i-1}\oplus T^{A_i}$, where $T_{i-1}$ is the $(i-1{)}^{th}$ attachment tree. Here one of the penultimate vertices of $T^{A_1}$, $i\ge 1$ is chosen for merging with the vertices of degree at least two of $T_{i-1}$, for $i\ge 1$. In this paper, we prove that for every $i\ge 1$, the $i$th attachment tree $T_i$ is graceful and admits a $\beta$-valuation. Thus it follows that the famous graceful tree conjecture is true for this infinite class of $(i^{th})$ attachment trees $T_i^{\prime }s$, for all $i\ge 1$. Due to the results of Rosa $[21]$ and El-Zanati et al. $[5]$ the complete graphs $K_{2cm+1}$ and complete bipartite graphs $K_{qm,pm}$, for $c,p,m,q\ge 1$ can be decomposed into copies of $i$th attachment tree $T_i$, for all $i\ge 1$, where $m$ is the size of such $i$th attachment tree $T_i$.

A packing of $K_n$ with copies of $C_4$ (the cycle of length $4$), is an ordered triple $(V,\mathcal{C},L)$, where $V$ is the vertex set of the complete graph $K_n$, $C$ is a collection of edge-disjoint copies of $C_4$, and $L$ is the set of edges not belonging to a block of $\mathcal{C}$. The number $n$ is called the order of the packing and the set of unused edges $L$ is called the leave. If $C$ is as large as possible, then $(V,\mathcal{C},L)$ is called a maximum packing MPC$(n,4,1)$. We say that an handcuffed design $H(v,k,1)$ $(W,P)$ is embedded into an MPC$(n,4,1)$ $(V,C,L)$ if $W\subseteq V$ and there is an injective mapping $f:\mathcal{P}\to \mathcal{C}$ such that $P$ is a subgraph of $f(P)$ for every $P\in \mathcal{P}$. Let $\mathcal{S}\mathcal{H}(n,4,k)$ denote the set of the integers $v$ such that there exists an MPC$(n,4,1)$ which embeds an $H(v,k,1)$. If $n\equiv 1(\mathrm{mod}8)$ then an MPC$(n,4,1)$ coincides with a $4$-cycle system of order $n$ and $\mathcal{S}\mathcal{H}(n,4,k)$ is found by Milici and Quattrocchi, Discrete Math., $174(1997)$.

The aim of the present paper is to determine $\mathcal{S}\mathcal{H}(n,4,k)$ for every integer $n\not\equiv 1(\mathrm{mod}8)$, $n\ge 4$.