Connected k-dominating graphs

For a graph $G=(V,E)$, the $k$-dominating graph $D_k\left(G\right)$ of $G$ has vertices corresponding to the dominating sets of $G$ having cardinality at most $k$, where two vertices of $D_k\left(G\right)$ are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote the domination and upper domination numbers of $G$ by $\gamma \left(G\right)$ and $\Gamma \left(G\right)$, respectively, and the smallest integer $\epsilon$ for which $D_k\left(G\right)$ is connected for all $k\ge \epsilon$ by $d_0\left(G\right)$. It is known that $\Gamma \left(G\right)+1\le d_0\left(G\right)\le \left|V\right|$, but constructing a graph $G$ such that $d_0\left(G\right)>\Gamma \left(G\right)+1$ appears to be difficult.We present two related constructions. The first construction shows that for each integer $k\ge 3$ and each integer $r$ such that $1\le r\le k?1$, there exists a graph $G_{k,r}$ such that $\Gamma \left(G_{k,r}\right)=k$, $\gamma \left(G_{k,r}\right)=r+1$ and $d_0\left(G_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)?1$. The second construction shows that for each integer $k\ge 3$ and each integer $r$ such that $1\le r\le k?1$, there exists a graph $Q_{k,r}$ such that $\Gamma \left(Q_{k,r}\right)=k$, $\gamma \left(Q_{k,r}\right)=r$ and $d_0\left(Q_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)$.     

The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials $F_{m,n,1}\left(x\right)$ and $F_{m,n,2}\left(x\right)$. Then, he defined $A_m(n,k)$ and $B_m(n,k)$ to be the polynomials satisfying $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{A}_m(n,k)x^{n?k}{(x+1)}^k$ and $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{B}_m(n,k)x^{n?k}{(x+1)}^k$. In this paper, we give a combinatorial interpretation of the coefficients of $A_{m+1}(n,k)$ and prove a symmetry of the coefficients, i.e., $\left[m^s\right]A_{m+1}(n,k)=\left[m^{n?s}\right]A_{m+1}(n,n?k)$. We give a combinatorial interpretation of $B_{m+1}(n,k)$ and prove that $B_{m+1}(n,n?1)$ is a polynomial in $m$ with non-negative integer coefficients. We also prove that if $n\ge 6$ then all coefficients of $B_{m+1}(n,n?2)$ except the coefficient of $m^{n?1}$ are non-negative integers. For all $n$, the coefficient of $m^{n?1}$ in $B_{m+1}(n,n?2)$ is $?(n?1)$, and when $n\le 5$ some other coefficients of $B_{m+1}(n,n?2)$ are also negative.     

Context-free grammars,generating functions and combinatorial arrays

Let ${\left[R_{n,k}\right]}_{n,k\ge 0}$ be an array of nonnegative numbers satisfying the recurrence relation $R_{n,k}=(a_{1}n+a_{2}k+a_3)R_{n-1,k}+(b_{1}n+b_{2}k+b_3)R_{n-1,k-1}+(c_{1}n+c_{2}k+c_3)R_{n-1,k-2}$ with $R_{0,0}=1$ and $R_{n,k}=0$ unless $0\le k\le n$. In this paper, we first prove that the array ${\left[R_{n,k}\right]}_{n,k\ge 0}$ can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of ${\left[R_{n,k}\right]}_{n,k\ge 0}$. Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type $B$, and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong $q$-log-convexity of some generating functions.     

Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

Let ${\left(Y_t\right)}_{t\ge 0}$ be an ergodic diffusion with invariant distribution $\nu$. Consider the empirical measure ${\nu }_n≔{\left({\sum }_{k=1}^n{\gamma }_k\right)}^{-1}{\sum }_{k=1}^n{\gamma }_k{\delta }_{X_{k-1}}$ where ${\left(X_k\right)}_{k\ge 0}$ is an Euler scheme with decreasing steps ${\left({\gamma }_k\right)}_{k\ge 0}$ which approximates ${\left(Y_t\right)}_{t\ge 0}$. Given a test function $f$, we obtain sharp concentration inequalities for ${\nu }_n\left(f\right)-\nu \left(f\right)$ which improve the results in Honoré et al. (2019). Our hypotheses on the test function $f$ cover many real applications: either $f$ is supposed to be a coboundary of the infinitesimal generator of the diffusion, or $f$ is supposed to be Lipschitz.     

Complete graphs and complete bipartite graphs without rainbow path

Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs $G$ and $H$, the $k$-colored Gallai–Ramsey number $g{r}_k(G:H)$ is defined to be the minimum integer $n$ such that $\left(\genfrac{}{}{0ex}{}{n}{2}\right)\ge k$ and for every $N\ge n$, every rainbow $G$-free coloring (using all $k$ colors) of the complete graph $K_N$ contains a monochromatic copy of $H$. In this paper, we first provide some exact values and bounds of $g{r}_k(P_5:K_t)$. Moreover, we define the $k$-colored bipartite Gallai–Ramsey number $bg{r}_k(G:H)$ as the minimum integer $n$ such that $n^2\ge k$ and for every $N\ge n$, every rainbow $G$-free coloring (using all $k$ colors) of the complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. Furthermore, we describe the structures of complete bipartite graph $K_{n,n}$ with no rainbow $P_4$ and $P_5$, respectively. Finally, we find the exact values of $bg{r}_k(P_4:K_{s,t})$ ($1\le s\le t$), $bg{r}_k(P_4:F)$ (where $F$ is a subgraph of $K_{s,t}$), $bg{r}_k(P_5:K_{1,t})$ and $bg{r}_k(P_5:K_{2,t})$ by using the structural results.     

On the largest element in D(n)-quadruples

Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\dots ,a_m\}$ such that $a_i{a}_j+n$ is a perfect square for all $1\le i<j\le m$ is called a $D\left(n\right)$-$m$-tuple. In this paper, we consider the question, for a given integer $n$ which is not a perfect square, how large and how small can be the largest element in a $D\left(n\right)$-quadruple. We construct families of $D\left(n\right)$-quadruples in which the largest element is of order of magnitude ${\left|n\right|}^3$, resp. ${\left|n\right|}^{2∕5}$.     

Vertex partitions of (C3,C4,C6)-free planar graphs

A graph is $(k_1,k_2)$-colorable if it admits a vertex partition into a graph with maximum degree at most $k_1$ and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable. We also show that deciding whether a $(C_3,C_4,C_6)$-free planar graph is $(0,3)$-colorable is NP-complete.     

Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.     

Permutations with small maximal k-consecutive sums

Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $({\pi }_1,\dots ,{\pi }_n)$ of integers $1,\dots ,n$, we consider $k$-consecutive sums of $\pi$, i.e., $s_i?{\sum }_{j=0}^{k?1}{\pi }_{i+j}$ for $i=1,\dots ,n$, where we let ${\pi }_{n+j}={\pi }_j$. What we want to do in this paper is to know the exact value of $\mathsf{msum}(n,k)?min\left\{max\{s_i:i=1,\dots ,n\}?\frac{k(n+1)}{2}:\pi \in S_n\right\},$ where $S_n$ denotes the set of all permutations of $1,\dots ,n$. In this paper, we determine the exact values of $\mathsf{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary of the results, we obtain $\mathsf{msum}(n,3)$, $\mathsf{msum}(n,4)$ and $\mathsf{msum}(n,6)$ for any $n$.     

Construction and enumeration of left dihedral codes satisfying certain duality properties

Let ${\mathbb{F}}_q$ be the finite field of q elements and let $D_{2n}=〈x,y|x^n=1,y^2=1,yxy=x^{n?1}〉$ be the dihedral group of 2n elements. Left ideals of the group algebra ${\mathbb{F}}_q[D_{2n}]$ are known as left dihedral codes over ${\mathbb{F}}_q$ of length 2n, and abbreviated as left $D_{2n}$-codes. Let $\gcd (n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over ${\mathbb{F}}_q$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over ${\mathbb{F}}_q$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over ${\mathbb{F}}_q$, and present several numerical examples to illustrative our applications.     

The component counts of random functions

Consider functions $f:A\to A\cup C$, where A and C are disjoint finite sets. The weakly connected components of the digraph of such a function are cycles of rooted trees, as in random mappings, and isolated rooted trees. Let $n_1=|A|$ and $n_3=|C|$. When a function is chosen from all ${(n_1+n_3)}^{n_1}$ possibilities uniformly at random, then we find the following limiting behaviour as $n_1\to \infty$. If $n_3=o(n_1)$, then the size of the maximal mapping component goes to infinity almost surely; if $n_3～\gamma n_1$, $\gamma >0$ a constant, then process counting numbers of mapping components of different sizes converges; if $n_1=o(n_3)$, then the number of mapping components converges to 0 in probability. We get estimates on the size of the largest tree component which are of order $\log ?n_3$ when $n_3～\gamma n_1$ and constant when $n_3～n_1^{\alpha }$, $\alpha >1$. These results are similar to ones obtained previously for random injections, for which the weakly connected components are cycles and linear trees.     

Stable-like fluctuations of Biggins’ martingales

Let ${\left(W_n\left(\theta \right)\right)}_{n\in {\mathbb{N}}_0}$ be Biggins’ martingale associated with a supercritical branching random walk, and let $W\left(\theta \right)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1\left(\theta \right)$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$, then, as $n\to \infty$, there is weak convergence of the tail process ${(W\left(\theta \right)?W_{n?k}\left(\theta \right))}_{k\in {\mathbb{N}}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.     

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).