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.     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.     

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.     

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.     

On the boomerang uniformity of a class of permutation quadrinomials over finite fields

Let ${\mathbb{F}}_{2^n}$ be a finite field with $2^n$ elements and $f_{\underset{\_}{c}}(x)=c_0{x}^{2^m(2^k+1)}+c_1{x}^{2^{m+k}+1}+c_2{x}^{2^m+2^k}+c_3{x}^{2^k+1}\in {\mathbb{F}}_{2^n}[x]$, where n, m and k are positive integers with $n=2m$ and $\gcd ?(m,k)=e$. In this paper, motivated by a recent work of Li, Xiong and Zeng (Li et al. (2021) [12]), we further study the boomerang uniformity of $f_{\underset{\_}{c}}(x)$ by using similar ideas and carrying out particular techniques in solving equations over finite fields. As a consequence, we generalize Li, Xiong and Zeng's result from the case of m being odd and $e=1$ to that of both $m/e$ and $k/e$ being odd.     

Erdős–Gallai stability theorem for linear forests

The Erd?s–Gallai Theorem states that every graph of average degree more than $l?2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd?s–Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=\left({?}_{i=1}^k{P}_{2a_i}\right)?\left({?}_{i=1}^l{P}_{2b_i+1}\right)$ be $k+l$ disjoint paths of order $2a_1,\dots ,2a_k,2b_1+1,\dots ,2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\ge 2$. If the minimum degree $\delta \left(G\right)\ge {\sum }_{i=1}^k{a}_i+{\sum }_{i=1}^l{b}_i?1$, then $F?G$ except several classes of graphs for sufficiently large $n$, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.     

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 $d_0\left(Q_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)$.     

Sharp upper bounds of the spectral radius of a graph

Let $G$ be a simple connected graph with $n$ vertices and $m$ edges. The spectral radius $\rho \left(G\right)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of $\rho \left(G\right)$ which improves some known bounds: If $\frac{(k?2)(k?3)}{2}\le m?n\le \frac{k(k?3)}{2}$, where $k(3\le k\le n)$ is an integer, then $\rho \left(G\right)\le \sqrt{2m?n?k+\frac{5}{2}+\sqrt{2m?2n+\frac{9}{4}}}.$The equality holds if and only if $G$ is a complete graph $K_n$ or $K_4?e$, where $K_4?e$ is the graph obtained from $K_4$ by deleting some edge $e$.     

Decomposition of complete equipartite graphs into paths and cycles of length 2p

Let $P_k$ and $C_k$ respectively denote a path and a cycle on k vertices. In this paper, we give necessary and sufficient conditions for the existence of a complete $\{P_{2p+1},C_{2p}\}$-decomposition of even regular complete equipartite graphs for all prime p.