In this paper, we show the necessary and sufficient conditions for a complete graph on $n$ vertices with a hole of size $v$ ($K_n\setminus K_v$) to be decomposed into isomorphic copies of $K_3$ with a pendant edge.

Given a digraph (an undirected graph, resp.) $D$ and two positive integers $f(x),g(x)$ for every $x\in V(D)$, a subgraph $H$ of $D$ is called a $(g,f)$-factor if $g(x)\le d_H^{+}(x)=d_H^{-}(x)\le f(x)$ ($g(x)\le d_H(x)\le f(x)$, resp.) for every $x\in V(D)$. If $f(x)=g(x)=1$ for every $x$, then a connected $(g,f)$-factor is a hamiltonian cycle. The previous research related to the topic has been carried out either for $(g,f)$-factors (in general, disconnected) or for hamiltonian cycles separately, even though numerous similarities between them have been recently detected. Here we consider connected $(g,f)$-factors in digraphs and show that several results on hamiltonian digraphs, which are generalizations of tournaments, can be extended to connected $(g,f)$-factors. Applications of these results to supereulerian digraphs are also obtained.

Let $G$ be a group of permutations acting on an $7$-vertex set $V$, and $X$ and $Y$ be two simple graphs on $V$. We say that $X$ and $Y$ are $G$-isomorphic if $Y$ belongs to the orbit of $X$ under the action of $G$. One can naturally generalize the reconstruction problems so that when $G$ is $S_v$, the symmetric group, we have the usual reconstruction problems. In this paper, we study $G$-edge reconstructibility of graphs. We prove some old and new results on edge reconstruction and reconstruction from end vertex deleted subgraphs.

Frucht and Salinas [1] conjectured that $C(k)\cup P(n)$ ($n\ge 3$) is graceful if and only if $k+n\ge 7$. We prove that $C(2k)\cup P(n)$ is graceful for $n>k+1$ ($k\ge 3$).

For smaller cases we prove that $C(2k)\cup P(n)$ is graceful for $k=3,4,5,6;n\ge 2$.

We present necessary and sufficient conditions for the decomposition of the complete symmetric bipartite digraph into each of the orientations of a $4$-cycle (in the cases for which such decompositions are not already known). We use these results to find optimal packings of the complete symmetric digraph with each of the orientations of a $4$-cycle. Finally, we give necessary and sufficient conditions for the existence of a decomposition of the complete symmetric digraph on $v$ vertices with a hole of size $w$ into each of the orientations of a $4$-cycle.

The sum graph of a set $S$ of positive integers is the graph $G^{+}(S)$ having $S$ as its vertex set, with two distinct vertices adjacent whenever their sum is in $S$. If $S$ is allowed to be a subset of all integers, the graph so obtained is called an integral sum graph. The integral sum number of a given graph $G$ is the smallest number of isolated vertices which when added to $G$ result in an integral sum graph. In this paper, we find the integral sum numbers of caterpillars, cycles, wheels, and complete bipartite graphs.

Let $k$ Max MOLS$(n)$ denote a maximal set of $k$ mutually orthogonal Latin squares of order $n$, and let the parameter triple $(G,n,k)$ denote the existence of a $k$ Max MOLS$(n)$ constructed from orthogonal orthomorphisms of a group $G$ of order $x$. We identify all such parameter triples for all $G$ of order $\le 15$, and report the existence of $3$ Max MOLS$(n)$ for $n=15,16$ and $4$ Max MOLS$(n)$ for $n=12,16,24,28$. Our work shows that for $n\le 15$, all known parameter pairs $(n,k)$ for which there exists a $k$ Max MOLS$(n)$ can be attained by constructing maximal sets of MOLS from orthomorphisms of groups, except for $1$ Max MOLS$(n)$, $n=5,7,9,13$ and $2$ Max MOLS$(10)$.

An alternating circular list of distinct $r$-element subsets of some finite set $X$ and distinct $r$-partitions of type $\tau$ is said to be a $\tau$-loop if successive members of the list are orthogonal. We address the problem of finding complete $\tau$-loops including all $r$-element subsets of $X$, for any fixed $|X|$ and type $\tau$.

The general Randić index $w_{\alpha }(G)$ of a graph $G$ is the sum of the weights $(d_G(u)d_G(v){)}^{\alpha }$ of all edges $uv$ of $G$. We give bounds for $w_{-1}(T)$ when $T$ is a tree of order $n$. We also show that $li{m}_{n\to \mathrm{\infty }}f(n)/n$ exists, and give bounds for the limit, where $f(n)=max\{w_{-1}(T):T$ is a tree of order $n$}. Finally, we find the expected value and variance of $w_{\alpha }(T)$ for certain families of trees.

The Hermitean forms graphs Her$(n,s)$ are a series of linear distance-regular graphs. The graph Her$(2,3)$ has the coset graph of the shortened ternary Golay code as an antipodal distance-regular cover. We give a new construction for this linear $3$-cover of $Her\text{(}(2,3)$ and show that it is unique.

A cyclic triple, $(a,b,c)$, is defined to be the set $\{(a,b),(b,c),(c,a)\}$ of ordered pairs. A Mendelsohn triple system of order $v$, MTS$(v)$, is a pair $(M,\beta )$, where $M$ is a set of $v$ points and $\beta$ is a collection of cyclic triples of pairwise distinct points of $M$ such that any ordered pair of distinct points of $M$ is contained in precisely one cyclic triple of $\beta$. An antiautomorphism of a Mendelsohn triple system, $(M,\beta )$, is a permutation of $M$ which maps $\beta$ to ${\beta }^{-1}$, where ${\beta }^{-1}=\{(c,b,a)\mid (a,b,c)\in \beta \}$. In this paper, we give necessary and sufficient conditions for the existence of a Mendelsohn triple system of order $v$ admitting an antiautomorphism consisting of two cycles of equal length and having $0$ or $1$ fixed points.

Let $G$ be a finite group and let ${\nu }_i(G)$ denote the proportion of ordered pairs of $G$ that generate a subgroup of nilpotency class $i$. Various properties of the ${\nu }_i(G)$’s are established. In particular, it is shown that ${\nu }_i(G)=k_i|G|/|G{|}^2$ for some non-negative integer $k_i$ and that $\sum _{i=1}^{\mathrm{\infty }}{\nu }_i$ is either $1$ or at most $\frac{1}{2}$ for solvable groups.

Two combinatorial identities are proved:
(1) $H_n(\epsilon )=\frac{n+2}{3}{M}_n(\epsilon )$, where $H_n(\epsilon )$ denotes the total number of vertices in all the n-edged rooted planar Eulerian maps and $M_n(\epsilon )$ denotes the number of such maps.
(2) $H_n(\mathcal{L})=\frac{5n^2+13n+2}{2(4n+1)}{M}_n(\mathcal{L})$, where $H_n(\mathcal{L})$ and $M_n(\mathcal{L})$ are defined similarly for the class $\mathcal{L}$ of loopless maps.
Simple closed formulae for $M_n(\epsilon )$ and $M_n(\mathcal{L})$ are well known, and they correspond to equivalent binomial sum identities.

We derive the exact joint distribution and prove the asymptotic joint normality of the numbers of peaks, double rises, troughs, and double falls in a random permutation. A Chi-square randomness test, as a by-product, is also proposed.

For a graph $G$ with vertex set $V$, the total redundance, $\text{TR}(G)$, and efficiency, $\text{F}(G)$, are defined by the two expressions:
$\text{TR}(G)=min\{\sum _{v\in S}(1+\mathrm{deg}v):S\subseteq V\text{and}[N(x)\cap S]\ge 1\mathrm{\forall }x\in V\},$
$\text{F}(G)=max\{\sum _{v\in S}(1+\mathrm{deg}v):S\subseteq V\text{and}[N(x)\cap S]\le 1\mathrm{\forall }x\in V\}.$
That is, $\text{TR}$ measures the minimum possible amount of domination if every vertex is dominated at least once, and $\text{F}$ measures the maximum number of vertices that can be dominated if no vertex is dominated more than once.

We establish sharp upper and lower bounds on $\text{TR}(G)$ and $\text{F}(G)$ for general graphs $G$ and, in particular, for trees, and briefly consider related Nordhaus-Gaddum-type results.

In this paper, generalizations of edge-cordial labelings are introduced and studied for special classes of trees and graphs.

The total of $4079$ $2$-designs and two $3$-designs on $21$ points have been found. All these designs have the same group as an automorphism group. This group can be represented as the wreath product of $G$ and $H$, where $G$ denotes the subgroup of order 3 of $\text{PSL}(2,2)$ and $H$ denotes the transitive subgroup of order 21 of $\text{PSL}(3,2)$.

In particular, $1,20,101,93,173,824$ and $2867$ values of $A$ for $2$-$(21,k,\lambda )$ designs have been detected, where $k$ takes values from $4$ through $10$. Up to our knowledge, $2217$ of these $\lambda$-values are new ($14,76,65,122,587$, and $1353$, for $k$ equal to $5,6,\dots ,10$, respectively). By Alltop’s extension [4], $1353$ new $2$-$(21,10,A)$ designs can be extended to the same number of new $3$-$(22,11,\lambda )$ designs.

An extensive search with $t>2$ and $k<8$ has given only the $3$-$(21,6,216)$ design and the $3$-$(21,7,1260)$ design with the same automorphism group.

We give new sets of sequences with zero autocorrelation function and entries from the set $\{0,\pm a,\pm b,\pm c\}$ where $a,b$ and $c$ are commuting variables (which may also be set zero). Then we use these sequences to construct some new orthogonal designs.

We show the known necessary conditions for the existence of an OD$(28;s_1,s_2,s_3)$ plus the condition that $(s_1,s_2,s_3)\ne (1,5,20)$ are sufficient conditions for the existence of an OD$(28;s_1,s_2,s_3)$. We also show the known necessary conditions for the existence of an OD$(28;s_1,s_2,s_3)$ constructed using four circulant matrices are sufficient conditions for the existence of 4-NPAF$(s_1,s_2,s_3)$ of length 2 for all lengths $n\ge 7$.

We establish asymptotic existence results for OD$(4N;s_1,s_2)$ for $3\le s_1+s_2\le 28$. This leaves no cases undecided for $1\le s_1+s_2\le 28$. We show the known necessary conditions for the existence of an OD$(28;s_1,s_2)$ with $25\le s_1+s_2\le 28$, constructed using four circulant matrices, plus the condition that $(s_1,s_2)\ne (1,26),(2,25),(7,19),(8,19)$ or $(13,14)$, are sufficient conditions for the existence of 4-NPAF$(s_1,s_2)$ of length $n$ for all lengths $n\ge 7$.

Let $G=G(V,E)$ be a graph. A labeling of $G$ is a function $f:V\to \{0,1,\dots ,n\}$ such that for each edge $e=(u,v)\in E$, $f(e)=|f(u)–f(v)|$. Such a labeling is said to be $k$-equitable if it is a labeling of the vertices with the numbers $0$ through $k–1$ such that, if $v_i$ is the number of vertices labeled $i$, and $e^i$ is the number of edges labeled $i$, then $|v^i–v^j|<1$ and $|e^i–e^j|\le 1$ for all $i,j$. A graph is said to be $k$-equitable if it has a $k$-equitable labeling. In this paper, we characterize the $k$-equitability of complete bipartite graphs and consider the equitability of complete multipartite graphs.

Let $D$ be a connected symmetric digraph, $A$ a finite abelian group with some specified property and $g\in A$. We present a characterization for two $g$-cyclic $A$-covers of $D$ to be isomorphic with respect to a group $\mathrm{\Gamma }$ of automorphisms of $D$, for any $g$ of odd order. Furthermore, we consider the number of $\mathrm{\Gamma }$-isomorphism classes of $g$-cyclic $A$-covers of $D$ for an element $g$ of odd order. We enumerate the number of isomorphism classes of $g$-cyclic $Z_{p^n}$-covers of $D$ with respect to the trivial group of automorphisms of $D$, for any prime $p(>2)$, where ${\mathbb{Z}}_{p^n}$ is the cyclic group of order $p^n$. Finally, we count $\mathrm{\Gamma }$-isomorphism classes of cyclic $F_p$-covers of $D$.

We completely settle the existence problem for group divisible designs with first and second associates in which the block size is $3$, and with $m$ groups each of size $n$, where $n,m\ge 3$.

We give a new and simple proof for the cyclic group of line crossings on the $2-D$ torus.

An abdiff-tolerance competition graph, $G=(V,E)$, is a graph for which each vertex $i$ can be assigned a non-negative integer $t_i$; and at most $|V|$ subsets $S_j$ of $V$ can be found such that $xy\in E$ if and only if $x$ and $y$ lie in at least $|t_x–t_y|$ of the sets $S_j$. If $G$ is not an abdiff-tolerance competition graph, it still is possible to find $r>|V|$ subsets of $V$ having the above property. The integer $r–|V|$ is called the abdiff-tolerance competition number. This paper determines those complete bipartite graphs which are abdiff-tolerance competition graphs and finds an asymptotic value for the abdiff-tolerance competition number of $K_{l,n}$.