Size and structure of large (s,t)(s,t)-union intersecting families

A family \F of sets is said to be intersecting if any two sets in \F have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set $X$. An $(s,t)$-union intersecting family is a family of $k$-sets on an $n$-set $X$ such that for any $A_1,\ldots,A_{s+t}$ in this family, $\left(\cup_{i=1}^sA_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ Let \ell(\F) be the minimum number of sets in \F such that by removing them the resulting subfamily is intersecting. In this paper, for sufficiently large $n$, we characterize the size and structure of $(s,t)$-union intersecting families with maximum possible size and \ell(\F)\geq s+\beta. This allows us to find out the size and structure of some large and maximal $(s,t)$-union intersecting families. Our results are nontrivial extensions of some recent generalizations of the Erd{\H o}s-Ko-Rado theorem such as the Han and Kohayakawa theorem 2017 which finds the structure of the third largest intersecting family, the Kostochka and Mubayi theorem 2017, and the more recent Kupavskii's theorem 2018 whose both results determine the size and structure of the $i$th largest intersecting family of $k$-sets for $i\leq k+1$. In particular, we prove that a Hilton-Milner-type stability theorem holds for $(1,t)$-union intersecting families, that indeed, confirms a conjecture of Alishahi and Taherkhani 2018. We extend our results to $K_{s_1,\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In fact, when $n$ is sufficiently large, we characterize the size and structure of large and maximal $K_{s_1,\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In particular, when $s_1=\cdots=s_{r+1}=1$ our result provides some stability results related to the famous Erd{\H o}s matching conjecture

r-Dynamic Chromatic Number of Graphs

An $r$-dynamic $k$-coloring of a graph $G$ is a proper vertex $k$-coloring such that the neighbors of any vertex $v$ receive at least $\min\{r,{\rm deg}(v)\}$ different colors. The $r$-dynamic chromatic number of $G$, $\chi_r(G)$, is defined as the smallest $k$ such that $G$ admits an $r$-dynamic $k$-coloring. In this paper we introduce an upper bound for $\chi_r(G)$ in terms of $r$, chromatic number, maximum degree and minimum degree. In 2001, Montgomery \cite{MR2702379} conjectured that, for a $d$-regular graph $G$, $\chi_2(G)-\chi(G)\leq 2$. In this regard, for a $d$-regular graph $G$, we present two upper bounds for $\chi_2(G)-\chi(G)$, one of them, $\lceil 5.437\log d+2.721\rceil$, is an improvement of the bound $14.06\log d +1$, proved by Alishahi (2011) \cite{MR2746973}. Also, we give an upper bound for $\chi_2(G)$ in terms of chromatic number, maximum degree and minimum degree.Comment: 9 page

On Coloring Properties of Graph Powers

This paper studies some coloring properties of graph powers. We show that $\chi_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)\chi_c(G)}{(s-r)\chi_c(G)+2r+1}$ provided that $\chi_c(G^{^{\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that if ${2r+1 \over 2s+1} \leq {\chi_c(G) \over 3(\chi_c(G)-2)}$, then $\chi_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)\chi_c(G)}{(s-r)\chi_c(G)+2r+1}$. In particular, $\chi_c(K_{3n+1}^{^{1\over3}})={9n+3\over 3n+2}$ and $K_{3n+1}^{^{1\over3}}$ has no subgraph with circular chromatic number equal to ${6n+1\over 2n+1}$. This provides a negative answer to a question asked in [Xuding Zhu, Circular chromatic number: a survey, Discrete Math., 229(1-3):371--410, 2001]. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that $\chi_f(G^{^{\frac{1}{2s+1}}})\leq \frac{(2s+1)\chi_f(G)}{s\chi_f(G)+1}$. Finally, we investigate the $n$th multichromatic number of subdivision graphs

A Note on Chromatic Sum

The chromatic sum $\Sigma(G)$ of a graph $G$ is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present $\Sigma(G)<\chi_f(G)|G|$ for every graph $G$

Extremal GG-free induced subgraphs of Kneser graphs

The Kneser graph ${\rm KG}_{n,k}$ is a graph whose vertex set is the family of all $k$-subsets of $[n]$ and two vertices are adjacent if their corresponding subsets are disjoint. The classical Erd\H{o}s-Ko-Rado theorem determines the cardinality and structure of a maximum induced $K_2$-free subgraph in ${\rm KG}_{n,k}$. As a generalization of the Erd\H{o}s-Ko-Rado theorem, Erd\H{o}s proposed a conjecture about the maximum order of an induced $K_{s+1}$-free subgraph of ${\rm KG}_{n,k}$. As the best known result concerning this conjecture, Frankl [Journal of Combinatorial Theory, Series A, 2013], when $n\geq(2s+1)k-s$, gave an affirmative answer to this conjecture and also determined the structure of such a subgraph. In this paper, generalizing the Erd\H{o}s-Ko-Rado theorem and the Erd{\H o}s matching conjecture, we consider the problem of determining the structure of a maximum family $\mathcal{A}$ for which ${\rm KG}_{n,k}[\mathcal{A}]$ has no subgraph isomorphic to a given graph $G$. In this regard, we determine the size and the structure of such a family provided that $n$ is sufficiently large with respect to $G$ and $k$. Furthermore, for the case $G=K_{1,t}$, we present a Hilton-Milner type theorem regarding above-mentioned problem, which specializes to an improvement of a result by Gerbner et al. [SIAM Journal on Discrete Mathematics, 2012].Comment: Minor change

Coloring Properties of Categorical Product of General Kneser Hypergraphs

More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be extremely better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture

On the random version of the Erd\H{o}s matching conjecture

The Kneser hypergraph ${\rm KG}^r_{n,k}$ is an $r$-uniform hypergraph with vertex set consisting of all $k$-subsets of $\{1,\ldots,n\}$ and any collection of $r$ vertices forms an edge if their corresponding $k$-sets are pairwise disjoint. The random Kneser hypergraph ${\rm KG}^r_{n,k}(p)$ is a spanning subhypergraph of ${\rm KG}^r_{n,k}$ in which each edge of ${\rm KG}^r_{n,k}$ is retained independently of each other with probability $p$. The independence number of random subgraphs of ${\rm KG}^2_{n,k}$ was recently addressed in a series of works by Bollob{\'a}s, Narayanan, and Raigorodskii (2016), Balogh, Bollob{\'a}s, and Narayanan (2015), Das and Tran (2016), and Devlin and Kahn (2016). It was proved that the random counterpart of the Erd\H{o}s-Ko-Rado theorem continues to be valid even for very small values of $p$. In this paper, generalizing this result, we will investigate the independence number of random Kneser hypergraphs ${\rm KG}^r_{n,k}(p)$. Broadly speaking, when $k$ is much smaller that $n$, we will prove that the random analogue of the Erd\H{o}s matching conjecture is true even for extremely small values of $p$.Comment: 11 page

A Catlin-type Theorem for Graph Partitioning Avoiding Prescribed Subgraphs

As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $\Delta(H)$, then $H$ has a vertex $\Delta(H)$-coloring such that one of the color classes is a maximum independent set. Let $G$ be a connected graph of order at least $2$. A $G$-free $k$-coloring of a graph $H$ is a partition of the vertex set of $H$ into $V_1,\ldots,V_k$ such that $H[V_i]$, the subgraph induced on $V_i$, does not contain any subgraph isomorphic to $G$. As a generalization of Catlin's theorem we show that a graph $H$ has a $G$-free $\lceil{\Delta(H)\over \delta(G)}\rceil$-coloring for which one of the color classes is a maximum $G$-free subset of $V(H)$ if $H$ satisfies the following conditions; (1) $H$ is not isomorphic to $G$ if $G$ is regular, (2) $H$ is not isomorphic to $K_{k\delta(G)+1}$ if $G \simeq K_{\delta(G)+1}$, and (3) $H$ is not an odd cycle if $G$ is isomorphic to $K_2$. Indeed, we show even more, by proving that if $G_1,\ldots,G_k$ are connected graphs with minimum degrees $d_1,\ldots,d_k$, respectively, and $\Delta(H)=\sum_{i=1}^{k}d_k$, then there is a partition of vertices of $H$ to $V_1,\ldots,V_k$ such that each $H[V_i]$ is $G_i$-free and moreover one of $V_i$s can be chosen in a way that $H[V_i]$ is a maximum $G_i$-free subset of $V(H)$ except either $k=1$ and $H$ is isomorphic to $G_1$, each $G_i$ is isomorphic to $K_{d_i+1}$ and $H$ is not isomorphic to $K_{\Delta(H)+1}$, or each $G_i$ is isomorphic to $K_{2}$ and $H$ is not an odd cycle.Comment: 8 page

Graph Powers and Graph Homomorphisms

In this paper we introduce a fractional power. Then, we present some properties of it. In this regard, we show that if G and H are two graphs and 1 ≤ 2r+

A Generalization of the Erd\"{o}s-Ko-Rado Theorem

In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we present an upper bound for their chromatic number