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 $e$.     

Panchromatic 3-colorings of random hypergraphs

The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let $H(n,k,p)$ denote the binomial model of a random $k$-uniform hypergraph on $n$ vertices. For given fixed $c>0$, $k⩾3$ and $p=cn∕\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, we prove that if $c<\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}-O\left({\left(\frac{\sqrt{3}}{2}\right)}^k\right)$then $H(n,k,p)$ admits a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$, but if $k$ is large enough and $c>\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}+O\left({\left(\frac{3}{4}\right)}^k\right)$then $H(n,k,p)$ does not admit a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$.     

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.     

Matching in 3-uniform hypergraphs

A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. We use $E_3(2d?1,n?2d+1)$ to denote the 3-uniform hypergraph whose vertex set can be partitioned into two vertex classes $V_1$ and $V_2$ of size $2d?1$ and $n?2d+1$, respectively, and whose edge set consists of all the triples containing at least two vertices of $V_1$. Let $H$ be a 3-uniform hypergraph of order $n\ge 13d$ with no isolated vertex and $\text{deg}\left(u\right)+\text{deg}\left(v\right)>2\left(\left(\genfrac{}{}{0ex}{}{n?1}{2}\right)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)\right)$ for any two adjacent vertices $u,v\in V\left(H\right)$. In this paper, we show that $H$ contains a matching of size $d$ if and only if $H$ is not a subgraph of $E_3(2d?1,n?2d+1)$. This result improves our previous one in Zhang and Lu (2018).     

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).     

Independence number of products of Kneser graphs

We study the independence number of a product of Kneser graph $K(n,k)$ with itself, where we consider all four standard graph products. The cases of the direct, the lexicographic and the strong product of Kneser graphs are not difficult (the formula for $\alpha (K(n,k)⊠K(n,k))$ is presented in this paper), while the case of the Cartesian product of Kneser graphs is much more involved. We establish a lower bound and an upper bound for the independence number of $K(n,2)\square K(n,2)$, which are asymptotically tending to $n^3∕3$ and $3n^3∕8$, respectively. The former is obtained by a construction, which differs from the standard diagonalization procedure, while for the upper bound the $\ell$-independence number of Kneser graphs can be applied. We also establish some constructions in odd graphs $K(2k+1,k)$, which give a lower bound for the 2-independence number of these graphs, and prove that two such constructions give the same lower bound as a previously known one. Finally, we consider the $s$-stable Kneser graphs $K{(ks+1,k)}_{s-stab}$, derive a formula for their $\ell$-independence number, and give the exact value of the independence number of the Cartesian square of $K{(ks+1,k)}_{s-stab}$.     

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 $n$.     

On the (3,1)-choosability of planar graphs without adjacent cycles of length 5,6,7

A $(k,d)$-list assignment $L$ of a graph $G$ is a mapping that assigns to each vertex $v$ a list $L\left(v\right)$ of at least $k$ colors satisfying $|L\left(x\right)\cap L\left(y\right)|\le d$ for each edge $xy$. A graph $G$ is $(k,d)$-choosable if there exists an $L$-coloring of $G$ for every $(k,d)$-list assignment $L$. This concept is also known as choosability with separation. In this paper, we prove that any planar graph $G$ is $(3,1)$-choosable if any $i$-cycle is not adjacent to a $j$-cycle, where $5\le i\le 6$ and $5\le j\le 7$.     

Remarks on the distribution of colors in Gallai colorings

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \cdots \ge e_k$ of positive integers is an $(n,k)$-sequence if ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i\le k$. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer $k\ge 3$ there exists an integer $g\left(k\right)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g\left(k\right)$. They showed that $g\left(3\right)=5,g\left(4\right)=8$ and $2k-2\le g\left(k\right)\le 8k^2+1$.We show that $g\left(5\right)=10$ and give almost matching lower and upper bounds for $g\left(k\right)$ by showing that with suitable constants $\alpha ,\beta >0$, $\frac{\alpha k^{1.5}}{lnk}\le g\left(k\right)\le \beta k^{1.5}$ for all sufficiently large $k$.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.     

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

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.