We define the $(i,j)$-liars’ domination number of $G$, denoted by $LR(i,j)(G)$, to be the minimum cardinality of a set $L\subseteq V(G)$ such that detection devices placed at the vertices in $L$ can precisely determine the set of intruder locations when there are between 1 and $i$ intruders and at most $j$ detection devices that might “lie”.

We also define the $X(c_1,c_2,\dots ,c_t,\dots )$-domination number, denoted by ${\gamma }_{X(c_1,c_2,\dots ,c_t,\dots )}(G)$, to be the minimum cardinality of a set $D\subseteq V(G)$ such that, if $S\subseteq V(G)$ with $|S|=k$, then $|(\bigcup _{v\in S}N[v])\cap D|\ge c_k$. Thus, $D$ dominates each set of $k$ vertices at least $c_k$ times making ${\gamma }_{X(c_1,c_2,\dots ,c_t,\dots )}(G)$ a set-sized dominating parameter. We consider the relations between these set-sized dominating parameters and the liars’ dominating parameters.

For Cauchy numbers of the first kind $\{a_n\}n\ge 0$ and Cauchy numbers of the second kind $\{b_n\}n\ge 0$, this paper focuses on the log-convexity of some sequences related to $\{a_n\}n\ge 0$ and $\{b_n\}n\ge 0$. For example, we discuss log-convexity of $\{n|a_n|–|a_{n+1}|\}n\ge 1$, $\{bn+1–n{b}_n\}n\ge 1$, $\{n|a_n|\}n\ge 1$, and $\{(n+1)b_n{\}}_{n\ge 0}$. In addition, we investigate log-balancedness of some sequences involving $a_n$ (or $b_n$).

Let $G$ be a graph. We define the distance $d$ pebbling number of $G$ to be the smallest number $s$ such that if $s$ pebbles are placed on the vertices of $G$, then there must exist a sequence of pebbling moves which takes a pebble to a vertex which is at a distance of at least $d$ from its starting point. In this article, we evaluate the distance $d$ pebbling numbers for a directed cycle graph with $n$ vertices.

Let $G$ be a $k$-connected ($k\ge 2$) graph of order $n$. If $\gamma (G^c)\ge n–k$, then $G$ is Hamiltonian or $K_k\vee K_{k+1}^c$, where $\gamma (G^c)$ is the domination number of the complement of the graph $G$.

An \emph{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f:V(D)\to \{0,1,2\}$ such that every vertex $v\in V(D)$ with $f(v)=0$ has at least two in-neighbors assigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. The weight of an Italian dominating function is the sum $\sum _{v\in V(D)}f(v)$, and the minimum weight of an Italian dominating function $f$ is the \emph{Italian domination number}, denoted by ${\gamma }_I(D)$. We initiate the study of the Italian domination number for digraphs, and we present different sharp bounds on ${\gamma }_I(D)$. In addition, we determine the Italian domination number of some classes of digraphs. As applications of the bounds and properties on the Italian domination number in digraphs, we give some new and some known results of the Italian domination number in graphs.

A hypergraph $H$ with vertex set $V$ and edge set $E$ is called bipartite if $V$ can be partitioned into two subsets $V_1$ and $V_2$ such that $e\cap V_1\ne \varphi$ and $e\cap V_2\ne \varphi$ for any $e\in E$. A bipartite self-complementary 3-uniform hypergraph $H$ with partition $(V_1,V_2)$ of a vertex set $V$ such that $|V_1|=m$ and $|V_2|=n$ exists if and only if either (i) $m=n$ or (ii) $m\ne n$ and either $m$ or $n$ is congruent to 0 modulo 4 or (iii) $m\ne n$ and both $m$ and $n$ are congruent to 1 or 2 modulo 4.

In this paper we prove that, there exists a regular bipartite self-complementary 3-uniform hypergraph $H(V_1,V_2)$ with $|V_1|=m,|V_2|=n,m+n>3$ if and only if $m=n$ and $n$ is congruent to 0 or 1 modulo 4. Further we prove that, there exists a quasi-regular bipartite self-complementary 3-uniform hypergraph $H(V_1,V_2)$ with $|V_1|=m,|V_2|=n,m+n>3$ if and only if either $m=3,n=4$ or $m=n$ and $n$ is congruent to 2 or 3 modulo 4.

Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G)\to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\in V(G)$ with degree greater than 1, the greatest common divisor of the set of labels in the neighborhood of $v$ is 1. In this paper, we introduce techniques for finding neighborhood-prime labelings based on the Hamiltonicity of the graph, by using conditions on possible degrees of vertices, and by examining a neighborhood graph. In particular, classes of graphs shown to be neighborhood-prime include all generalized Petersen graphs, grid graphs of any size, and lobsters given restrictions on the degree of the vertices. In addition, we show that almost all graphs and almost all regular graphs have neighborhood-prime, and we find all graphs of this type.

Let $A(n,d,w)$ denote the maximum size of a binary code with length $n$, minimum distance $d$, and constant weight $w$. The following lower bounds are here obtained in computer searches for codes with prescribed automorphisms: $A(16,4,6)\ge 624$, $A(19,4,8)\ge 4698$, $A(20,4,8)\ge 7830$, $A(21,4,6)\ge 2880$, $A(22,6,6)\ge 343$, $A(24,4,5)\ge 1920$, $A(24,6,9)\ge 3080$, $A(24,6,11)\ge 5376$, $A(24,6,12)\ge 5558$, $A(25,4,5)\ge 2380$, $A(25,6,10)\ge 6600$, $A(26,4,5)\ge 2816$, and $A(27,4,5)\ge 3456$.

For a finite simple graph $G$, say $G$ is of dimension $n$, and write $\text{dim}(G)=n$, if $n$ is the smallest integer such that $G$ can be represented as a unit-distance graph in ${\mathbb{R}}^n$. Define $G$ to be \emph{dimension-critical} if every proper subgraph of $G$ has dimension less than $G$. In this article, we determine exactly which complete multipartite graphs are dimension-critical. It is then shown that for each $n\ge 2$, there is an arbitrarily large dimension-critical graph $G$ with $\text{dim}(G)=n$. We close with a few observations and questions that may aid in future work.

Let $G$ be a simple and finite graph. A graph is said to be decomposed into subgraphs $H_1$ and $H_2$ which is denoted by $G=H_1\oplus H_2$, if $G$ is the edge disjoint union of $H_1$ and $H_2$. If $G=H_1\oplus H_2\oplus \cdots \oplus H_k$, where $H_1,H_2,\dots ,H_k$ are all isomorphic to $H$, then $G$ is said to be $H$-decomposable. Furthermore, if $H$ is a cycle of length $m$, then we say that $G$ is $C_m$-decomposable and this can be written as $C_m\mid G$. Where $G\times H$ denotes the tensor product of graphs $G$ and $H$, in this paper, we prove that the necessary conditions for the existence of $C_6$-decomposition of $K_m\times K_n$ are sufficient. Using these conditions it can be shown that every even regular complete multipartite graph $G$ is $C_6$-decomposable if the number of edges of $G$ is divisible by 6.

Let $F,G$ and $H$ be graphs. A $(G,H)$-decomposition of $F$ is a partition of the edge set of $F$ into copies of $G$ and copies of $H$ with at least one copy of $G$ and at least one copy of $H$. For $L\subseteq F$, a $(G,H)$-packing of $F$ with leave $L$ is a $(G,H)$-decomposition of $F–E(L)$. A $(G,H)$-packing of $F$ with the largest cardinality is a maximum $(G,H)$-packing. This paper gives the solution of finding the maximum $(C_k,S_k)$-packing of the crown $C_{n,n-1}$.

Rautenbach and Volkmann [Appl. Math. Lett. 20 (2007), 98–102] gave an upper bound for the $k$-domination number and $k$-tuple domination number of a graph. Hansberg and Volkmann, [Discrete Appl. Math. 157 (2009), 1634–1639] gave upper bounds for the $k$-domination number and Roman $k$-domination number of a graph. In this note, using the probabilistic method and the known Caro-Wei Theorem on the size of the independence number of a graph, we improve the above bounds on the $k$-domination number, the $k$-tuple domination number and the Roman $k$-domination number in a graph for any integer $k\ge 1$. The special case $k=1$ of our bounds improve the known bounds of Arnautov and Payan [V.I. Arnautov, Prikl. Mat. Programm. 11 (1974), 3–8 (in Russian); C. Payan, Cahiers Centre Études Recherche Opér. 17 (1975) 307–317] and Cockayne et al. [Discrete Math. 278 (2004), 11–22].

Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27–32), we prove ${\gamma }_{r2}(G)\le 2{\gamma }_r(G)$ for every graph $G$, where ${\gamma }_{r2}(G)$ and ${\gamma }_r(G)$ denote the 2-rainbow domination number and the weak Roman domination number of $G$, respectively. We characterize the extremal graphs for this inequality that are $\{K_4,K_4–e\}$-free, and show that the recognition of the $K_5$-free extremal graphs is NP-hard.

For a graph $H$, let ${\delta }_t(H)=min\{|\bigcup _{i=1}^t{N}_H(v_i)|:|v_1,\dots ,v_t|\text{ are }t\text{ vertices in }H\}$. We show that for a given number $ϵ$ and given integers $p\ge 2$ and $k\in \{2,3\}$, the family of $k$-connected Hamiltonian claw-free graphs $H$ of sufficiently large order $n$ with $\delta (H)\ge 3$ and ${\delta }_k(H)\ge t(n+ϵ)/p$ has a finite obstruction set in which each member is a $k$-edge-connected $K_3$-free graph of order at most $max\{p/t+2t,3p/t+2t–7\}$ and without spanning closed trails. We found the best possible values of $p$ and $ϵ$ for some $t\ge 2$ when the obstruction set is empty or has the Petersen graph only. In particular, we prove the following for such graphs $H$:

• (a) For $k=2$ and a given $t$ ($1\le t\le 4$), if ${\delta }_t(H)\ge \frac{n+1}{3}$ and $\delta (H)\ge 3$, then $H$ is Hamiltonian.
• (b) For $k=3$ and $t=3$, (i) if ${\delta }_3(H)\ge \frac{n+9}{10}$, then $H$ is Hamiltonian; (ii) if ${\delta }_2(H)\ge \frac{n+9}{10}$, then either $H$ is Hamiltonian, or $H$ can be characterized by the Petersen graph.
• (c) For $k=3$ and $t=3$, (i) if ${\delta }_3(H)\ge \frac{n+9}{8}$, then $H$ is Hamiltonian; (ii) if ${\delta }_3(H)\ge \frac{n+6}{9}$, then either $H$ is Hamiltonian, or $H$ can be characterized by the Petersen graph.

These bounds on ${\delta }_t(H)$ are sharp. Since the number of graphs of orders at most $max\{p/t+2t,3p/t+2t–7\}$ is finite for given $p$ and $t$, improvements to (a), (b), or (c) by increasing the value of $p$ are possible with the help of a computer.

Any dominating set of vertices in a triangle-free graph can be used to specify a graph coloring with at most one color class more than the number of vertices in the dominating set. This bound is sharp for many graphs. Properties of graphs for which this bound is achieved are presented.

A graph $G$ is quasi-claw-free if it satisfies the property: $d(x,y)=2$ implies there exists $u\in N(x)\cap N(y)$ such that $N[u]\subseteq N[x]\cup N[y]$. The matching number of a graph $G$ is the size of a maximum matching in the graph. In this note, we present a sufficient condition involving the matching number for the Hamiltonicity of quasi-claw-free graphs.

Let $S$ be an orthogonal polygon and let $A_1,\dots ,A_n$ represent pairwise disjoint sets, each the connected interior of an orthogonal polygon, $A_i\subseteq S,1\le i\le n$. Define $T=S\setminus (A_1\cup \dots \cup A_n)$. We have the following Krasnosel’skii-type result: Set $T$ is staircase star-shaped if and only if $S$ is staircase star-shaped and every $4n$ points of $T$ see via staircase paths in $T$ a common point of $\text{Ker }S$. Moreover, the proof offers a procedure to select a particular collection of $4n$ points of $T$ such that the subset of $\text{Ker }S$ seen by these $4n$ points is exactly $\text{Ker }T$. When $n=1$, the number 4 is best possible.

The $p$-competition graph $C_p(D)$ of a digraph $D=(V,A)$ is a graph with $V(C_p(D))=V(D)$, where an edge between distinct vertices $x$ and $y$ if and only if there exist $p$ distinct vertices $v_1,v_2,\dots ,v_p\in V$ such that $x\to v_i,y\to v_i$ are arcs of the digraph $D$ for each $i=1,2,\dots ,p$. In this paper, we prove that double stars $D{S}_m$ ($m\ge 2$) are $p$-competition graphs. We also show that full regular $m$-ary trees $T_{m,n}$ with height $n$ are $p$-competition graphs, where $p\le \frac{m–1}{2}$.

Let $G$ be a graph with at least half of the vertices having degree at least $k$. For a tree $T$ with $k$ edges, Loebl, Komlós, and Sós conjectured that $G$ contains $T$. It is known that if the length of a longest path in $T$ (i.e., the diameter of $T$) is at most 5, then $G$ contains $T$. Since $T$ is a bipartite graph, let $\ell$ be the number of vertices in the smaller (or equal) part. Clearly $1\le \ell \le \frac{1}{2}(k+1)$. In our main theorem, we prove that if $1\le \ell \le \frac{1}{6}k+1$, then the graph $G$ contains $T$. Notice that this includes certain trees of diameter up to $\frac{1}{3}k+2$.

If a tree $T$ consists of only a path and vertices that are connected to the path by an edge, then the tree $T$ is a caterpillar. Let $P$ be the path obtained from the caterpillar $T$ by removing each leaf of $T$, where $P=a_1,\dots ,a_r$. The path $P$ is the spine of the caterpillar $T$, and each vertex on the spine of $T$ with degree at least 3 in $T$ is a joint. It is known that the graph $G$ contains certain caterpillars having at most two joints. If only odd-indexed vertices on the spine $P$ are joints, then the caterpillar $T$ is an odd caterpillar. If the spine $P$ has at most $\lceil \frac{1}{2}k\rceil$ vertices, then $T$ is a short caterpillar. We prove that the graph $G$ contains every short, odd caterpillar with $k$ edges.

The decision problems of the existence of a Hamiltonian cycle or of a Hamiltonian path in a given graph, and of the existence of a truth assignment satisfying a given Boolean formula C, are well-known NP-complete problems. Here we study the problems of the uniqueness of a Hamiltonian cycle or path in an undirected, directed or oriented graph, and show that they have the same complexity, up to polynomials, as the problem U-SAT of the uniqueness of an assignment satisfying C. As a consequence, these Hamiltonian problems are NP-hard and belong to the class DP, like U-SAT.

A graph $G$ is $k$-frugal colorable if there exists a proper vertex coloring of $G$ such that every color appears at most $k–1$ times in the neighborhood of $v$. The $k$-frugal chromatic number, denoted by ${\chi }_k(G)$, is the smallest integer $l$ such that $G$ is $k$-frugal colorable with $l$ colors. A graph $G$ is $L$-list colorable if there exists a coloring $c$ of $G$ for a given list assignment $L=\{L(v):v\in V(G)\}$ such that $c(v)\in L(v)$ for all $v\in V(G)$. If $G$ is $k$-frugal $L$-colorable for any list assignment $L$ with $|L(v)|\ge l$ for all $v\in V(G)$, then $G$ is said to be $k$-frugal $l$-list-colorable. The smallest integer $l$ such that the graph $G$ is $k$-frugal $l$-list-colorable is called the $k$-frugal list chromatic number, denoted by ${\text{ch}}_k(G)$. It is clear that ${\text{ch}}_k(G)\ge \lceil \frac{\mathrm{\Delta }(G)}{k–1}\rceil +1$ for any graph $G$ with maximum degree $\mathrm{\Delta }(G)$. In this paper, we prove that for any integer $k\ge 4$, if $G$ is a planar graph with maximum degree $\mathrm{\Delta }(G)\ge 13k–11$ and girth $g\ge 6$, then ${\text{ch}}_k(G)=\lceil \frac{\mathrm{\Delta }(G)}{k–1}\rceil +1;$ and if $G$ is a planar graph with girth $g\ge 6$, then ${\text{ch}}_k(G)\le \lceil \frac{\mathrm{\Delta }(G)}{k–1}\rceil +2.$

In 1987, Alavi, Boals, Chartrand, Erdös, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). In previous papers, we showed that all tournaments of order congruent to 1, 2, or 3 mod 6 have an ASD. In this paper, we will consider the case where the tournament has order congruent to 5 mod 6.

An $H$-decomposition of a graph $G$ is a partition of the edges of $G$ into copies isomorphic to $H$. When the decomposition is not feasible, one looks for the best possible by minimizing: the number of unused edges (leave of a packing), or the number of reused edges (padding of a covering). We consider the $H$-decomposition, packing, and covering of the complete graphs and complete bipartite graphs, where $H$ is a 4-cycle with three pendant edges.

We introduce a new bivariate polynomial
$$J(G;x,y):=\sum _{W\subseteq V(G)}{x}^{|W|}{y}^{|N[W]|–|W|}$$
which contains the standard domination polynomial of the graph $G$ in two different ways. We build methods for efficient calculation of this polynomial and prove that there are still some families of graphs which have the same bivariate polynomial.

Let $G$ be a $(p,q)$ graph. Let $f:V(G)\to \{1,2,\dots ,k\}$ be a map where $k$ is an integer $2\le k\le p$. For each edge $uv$, assign the label $|f(u)–f(v)|$. $f$ is called $k$-difference cordial labeling of $G$ if $|v_f(i)–v_f(j)|\le 1$ and $|e_f(0)–e_f(1)|\le 1$, where $v_f(x)$ denotes the number of vertices labeled with $x$, $e_f(1)$ and $e_f(0)$ respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a $k$-difference cordial labeling is called a $k$-difference cordial graph. In this paper, we investigate 3-difference cordial labeling behavior of slanting ladder, book with triangular pages, middle graph of a path, shadow graph of a path, triangular ladder, and the armed crown.

In this paper, we consider the sequences $\{F(n,k){\}}_{n\ge k}$ ($k\ge 1$) defined by$F(n,k)=(n–2)F(n–1,k)+F(n–1,k–1),F(n,1)=\frac{n!}{2},F(n,n)=1.$ We mainly study the log-convexity of $\{F(n,k)\}n\ge k$ ($k\ge 1$) when $k$ is fixed. We prove that $\{F(n,3)\}n\ge 3,\{F(n,4)\}n\ge 5,$ and $\{F(n,5)\}n\ge 6$ are log-convex. In addition, we discuss the log-behavior of some sequences related to $F(n,k)$.
\end{abstract}

Let $G=C_n\oplus C_n$ with $n\ge 3$ and $S$ be a sequence with elements of $G$. Let $\mathrm{\Sigma }(S)\subseteq G$ denote the set of group elements which can be expressed as a sum of a nonempty subsequence of $S$. In this note, we show that if $S$ contains $2n–3$ elements of $G$, then either $0\in \mathrm{\Sigma }(S)$ or $|\mathrm{\Sigma }(S)|\ge n^2–n–1$. Moreover, we determine the structures of the sequence $S$ over $G$ with length $|S|=2n–3$ such that $0\notin \mathrm{\Sigma }(S)$ and $|\mathrm{\Sigma }(S)|=n^2–n–1$.