In considering packing three copies of a tree into a complete bipartite graph, H. Wang (2009) gives a conjecture: For each tree $T$ of order $n$ and each integer $k\ge 2$, there is a $k$-packing of $T$ in a complete bipartite graph $B_{n+k-1}$ whose order is $n+k-1$. We prove the conjecture is true for $k=4$.

We consider the hexagonal circle packing with radius $1/2$ and perturb it by letting the circles move as independent Brownian motions for time $t$. It is shown that, for large enough $t$, if ${𝛱}_t$ is the point process given by the center of the circles at time $t$, then, as $t\to \infty$, the critical radius for circles centered at ${𝛱}_t$ to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly...

We prove three theorems on linear operators $T_{\tau ,p}:H_p\left(ℬ\right)\to H_p$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{\tau ,p}$ to be continuous for 0 < p < ∞.

We deal with the so-called Ahlfors regular sets (also known as $s$-regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit ${lim}_{\epsilon \to 0+}{\epsilon }^{s}N(\epsilon ,K)$ exist, where $K$ is an $s$-regular set and $N(\epsilon ,K)$ is for instance the $\epsilon$-packing number of $K$?

Let g be a doubling gauge. We consider the packing measure ${}^g$ and the packing premeasure ${₀}^g$ in a metric space X. We first show that if ${₀}^g\left(X\right)$ is finite, then as a function of X, ${₀}^g$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $\lambda sup{₀}^g\left(F\right){\le }^g\left(B\right)\le sup{₀}^g\left(F\right)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite ${₀}^g$-premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling...

For a graph $G=(V,E)$, a double Roman dominating function is a function $f:V\to \{0,1,2,3\}$ having the property that if $f\left(v\right)=0$, then the vertex $v$ must have at least two neighbors assigned $2$ under $f$ or one neighbor with $f\left(w\right)=3$, and if $f\left(v\right)=1$, then the vertex $v$ must have at least one neighbor with $f\left(w\right)\ge 2$. The weight of a double Roman dominating function $f$ is the sum $f\left(V\right)={\sum }_{v\in V}f\left(v\right)$. The minimum weight of a double Roman dominating function on $G$ is called the double Roman domination number of $G$ and is denoted by ${\gamma }_{\mathrm{dR}}\left(G\right)$. In this paper, we establish a new...

Let $G$ be a finite group and construct a graph $\Delta \left(G\right)$ by taking $G\setminus \left\{1\right\}$ as the vertex set of $\Delta \left(G\right)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K\left(G\right)$ be the set consisting of the universal vertices of $\Delta \left(G\right)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient condition for $K\left(G\right)$ to be nontrivial. We also develop a connection between $\Delta \left(G\right)$ and $K\left(G\right)$ when $\left|G\right|$ is divisible by two distinct primes and the diameter of $\Delta \left(G\right)$ is 2.

Let $ex(n,G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_i$ denote a path consisting of $i$ vertices and let $m{P}_i$ denote $m$ disjoint copies of $P_i$. In this paper we count $ex(n,3P_4)$.

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G{\cong }^2{G}_2\left(3^{2n+1}\right)$ if and only if $𝒮\left(G\right)\cong 𝒮{(}^2{G}_2\left(3^{2n+1}\right))$. As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group ${}^2{G}_2\left(3^{2n+1}\right)$.

For a fixed positive integer $k$ and $G=(V,E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in V$ has at least $k$ neighbors in $S$. The minimum size of a total $k$-dominating set for $G$ is called the total $k$-domination number of $G$, denoted by ${\gamma }_{kt}\left(G\right)$. The total $k$-domination problem is to determine a minimum total $k$-dominating set of $G$. Since the exact problem is in general quite difficult...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\lim {\sup }_{n\to \infty }|A\cap {\left[k\right]}^n|/k^n>0$ there exist a word $c$ over $k$ and a sequence $\left(w_n\right)$ of left variable words over $k$ such that the set $c\cup \{c^{}{w}_0{\left(a_0\right)}^{}..{.}^{}{w}_n\left(a_n\right):n\in \mathbb{N}\text{and}{a}_0,...,a_n\in \left[k\right]\}$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

Observations are made on a point process $𝛯$ in ${ℝ}^d$ in a window $Q_{\lambda }$ of volume $\lambda$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,𝛯)$, is a function of the points within a random distance of $x$. When the input $𝛯$ is a Poisson or binomial point process, the large $\lambda$ limit theory for the total score ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian...

Let $G$ be a simple connected graph with vertex set $V\left(G\right)=\{v_1,v_2,\cdots ,v_n\}$ and edge set $E\left(G\right)$, and let $d_{v_i}$ be the degree of the vertex $v_i$. Let $D\left(G\right)$ be the distance matrix and let $T_r\left(G\right)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_{\alpha }\left(G\right)=\alpha T_r\left(G\right)+(1-\alpha )D\left(G\right)$, where $0\le \alpha \le 1$. Let ${\lambda }_1\left(D_{\alpha }\left(G\right)\right)\ge {\lambda }_2\left(D_{\alpha }\left(G\right)\right)\ge ...\ge {\lambda }_n\left(D_{\alpha }\left(G\right)\right)$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\le k\le n$. We denote by $S_k\left(D_{\alpha }\left(G\right)\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)+{\lambda }_2\left(D_{\alpha }\left(G\right)\right)+...+{\lambda }_k\left(D_{\alpha }\left(G\right)\right)$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S\left(G\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)-{\lambda }_n\left(D_{\alpha }\left(G\right)\right)$....

A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $\mathrm{sat}(n,H)$, is the minimum number of edges of a graph in the set of all $H$-saturated graphs of order $n$. We determine the saturation number $\mathrm{sat}(n,P_6+t{P}_2)$ for $n\ge \frac{10}{3}t+10$ and characterize the extremal graphs for $n>\frac{10}{3}t+20$.

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$. Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_1$ and $e_2$ of $G$, there is an even factor containing $e_1$ and $e_2$...

The line graph of a graph $G$, denoted by $L\left(G\right)$, has $E\left(G\right)$ as its vertex set, where two vertices in $L\left(G\right)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$, define ${\overline{\sigma }}_2\left(H\right)=min\{d\left(u\right)+d\left(v\right):uv\in E\left(H\right)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with $\delta \left(H\right)\ge 3$. We show that, if ${\overline{\sigma }}_2\left(H\right)\ge \frac{1}{7}(2n-5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $\mathrm{cl}\left(H\right)=L\left(G\right)$, where $G$ is an essentially $2$-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...