Analytic aspects of Delannoy numbers

The Delannoy numbers $d(n,k)$ count the number of lattice paths from $(0,0)$ to $(n?k,k)$ using steps $(1,0),(0,1)$ and $(1,1)$. We show that the zeros of all Delannoy polynomials $d_n\left(x\right)={\sum }_{k=0}^{n}d(n,k)x^k$ are in the open interval $(?3?2\sqrt{2},?3+2\sqrt{2})$ and are dense in the corresponding closed interval. We also show that the Delannoy numbers $d(n,k)$ are asymptotically normal (by central and local limit theorems).     

Clique number of Xor products of Kneser graphs

In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in “A general 2-part Erd?s-Ko-Rado theorem” by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic $KG(N,k)$ Kneser graphs. Denote this number with $f(k,N)$. We give lower and upper bounds on $f(k,N)$, and we solve the problem up to a constant deviation depending only on k, and find the exact value for $f(2,N)$ if N is large enough. Also we compute that $f(k,k^2)$ is asymptotically equivalent to $k^2$.     

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.     

Fractional chromatic numbers of tensor products of three graphs

The tensor product $(G_1,G_2,G_3)$ of graphs $G_1$, $G_2$ and $G_3$ is defined by $V(G_1,G_2,G_3)=V\left(G_1\right)\times V\left(G_2\right)\times V\left(G_3\right)$and $E(G_1,G_2,G_3)=\left\{\left((u_1,u_2,u_3),(v_1,v_2,v_3)\right):\left|\{i:(u_i,v_i)\in E\left(G_i\right)\}\right|\ge 2\right\}.$Let ${\chi }_f\left(G\right)$ be the fractional chromatic number of a graph $G$. In this paper, we prove that if one of the three graphs $G_1$, $G_2$ and $G_3$ is a circular clique, ${\chi }_f(G_1,G_2,G_3)=min\{{\chi }_f\left(G_1\right){\chi }_f\left(G_2\right),{\chi }_f\left(G_1\right){\chi }_f\left(G_3\right),{\chi }_f\left(G_2\right){\chi }_f\left(G_3\right)\}.$     

H-free subgraphs of dense graphs maximizing the number of cliques and their blow-ups

We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $???(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from which one can delete an edge and reduce the chromatic number, and for every graph $G$ on $n>n_0\left(H\right)$ vertices in which all degrees are at least $(1??)n$, any subgraph of $G$ which is $H$-free and contains the maximum number of copies of the complete graph $K_m$ is $(k?1)$-colorable.We also consider several extensions for the case of a general forbidden graph $H$ of a given chromatic number, and for subgraphs maximizing the number of copies of balanced blowups of complete graphs.     

Ramsey number of multiple copies of stars

Let G and H be simple graphs. The Ramsey number $r(G,H)$ is the minimum integer N such that any red-blue-coloring of edges of $K_N$ contains either a red copy of G or a blue copy of H. Let $m{K}_{1,t}$ denote m vertex-disjoint copies of $K_{1,t}$. A lower bound is that $r(m{K}_{1,t},n{K}_{1,s})\ge m(t+1)+n?1$. Burr, Erd?s and Spencer proved that this bound is indeed the Ramsey number $r(m{K}_{1,t},n{K}_{1,s})$ for $t=s=3$, $m\ge 2$ and $m\ge n$. In this paper, we show that this bound is the Ramsey number $r(m{K}_{1,t},n{K}_{1,s})$ for $t\ge s=3,m\ge 2$ and $m\ge n$. We also show that this bound is the Ramsey number $r(m{K}_{1,t},n{K}_{1,s})$ for $s\ge 4,t>\frac{s(s?1)}{2}$ and $m>n$.     

The graph Ramsey number R(Fℓ,K6)

For a given pair of two graphs $(F,H)$, let $R(F,H)$ be the smallest positive integer $r$ such that for any graph $G$ of order $r$, either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that $R(F_{\ell },K_n)=2\ell (n-1)+1$for $\ell \ge n\ge 3$, where $F_{\ell }$ is the join $K_1+\ell K_2$ of $K_1$ and $\ell K_2$. In this paper, we prove that this conjecture is true for the case $n=6$.     

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).     

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.     

A characterization of graphs with given maximum degree and smallest possible matching number: II

Let ${\alpha }^{\prime }(G)$ be the matching number of a graph G. A characterization of the graphs with given maximum odd degree and smallest possible matching number is given by Henning and Shozi (2021) [13]. In this paper we complete our study by giving a characterization of the graphs with given maximum even degree and smallest possible matching number. In 2018 Henning and Yeo [10] proved that if G is a connected graph of order n, size m and maximum degree k where $k\ge 4$ is even, then ${\alpha }^{\prime }(G)\ge \frac{n}{k(k+1)}+\frac{m}{k+1}?\frac{1}{k(k+1)}$, unless G is k-regular and $n\in \{k+1,k+3\}$. In this paper, we give a complete characterization of the graphs that achieve equality in this bound when the maximum degree k is even, thereby completing our study of graphs with given maximum degree and smallest possible matching number.     

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 $e$.     

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.