On Determining Number of Kneser Graphs

Electronic Notes in Discrete Mathematics, 2016

The achromatic number α of a graph is the largest number of colors that can be assigned to its vertices such that adjacent vertices have different color and every pair of different colors appears on the end vertices of some edge. We estimate the achromatic number of Kneser graphs K(n, k) and determine α(K(n, k)) for some values of n and k. Furthermore, we study the achromatic number of some geometric type Kneser graphs.

Discrete Applied Mathematics, 2011

Let c be a proper k-coloring of a connected graph G and Π = (C 1 , C 2 , . . . , C k ) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c

In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{\v{r}}{\'{\i}}{\v{z}} bound. Next, we introduce multiple Kneser hypergraphs and we specify the chromatic number of some multiple Kneser hypergraphs. For a vector of positive integers $\vec{s}=(s_1,s_2,\ldots,s_m)$ and a partition $\pi=(P_1,P_2,\ldots,P_m)$ of $\{1,2,\ldots,n\}$, the multiple Kneser hypergraph ${\rm KG}^r(\pi; \vec{s};k)$ is a hypergraph with the vertex set $$V=\left\{A:\ A\subseteq P_1\cup P_2\cup\cdots \cup P_m,\ |A|=k, \forall 1\leq i\leq m;\ |A\cap P_i|\leq s_i\right\}$$ whose edge set is consist of any $r$ pairwise disjoint vertices. We determine the chromatic number of multiple Kneser hypergraphs provided that $r=2$ or for any $1\leq i\leq m$, we have $|P_i|\leq 2s_i$. A subset $S \subseteq [n]$ is almost $s$-stable if for any two distinct elements $i,j\in S$, we have $|i-j|\geq s$. The almost $s$-stable Kneser h...

Computational Geometry, 2005

We estimate the chromatic number of graphs whose vertex set is the set of edges of a complete geometric graph on n points, and adjacency is defined in terms of geometric disjointness or geometric intersection.

arXiv: Combinatorics, 2014

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels, required for a graph $G$ to admit a weak IASI. In this paper, we study the sparing number of certain graphs and the relation of sparing number with some other parameters like matching number, chromatic number, covering number, independence number etc.

Let $G$ be a graph of order $n$. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$. The minimum value of $ f(V(G))$, taken over all good functions is denoted by $ \gamma(G) $. A function $f:V(G)\rightarrow \{-1,1\}$ is called an excellent function, if $f(N[v])\geq1$, for each $v\in V(G)$, where $ N[v]=\{v\} \cup N(v) $. Define $\overline{\gamma}(G)=min\{f(V(G))\}$, where $f$ is an excellent function for $G$. In this paper, we present several lower and upper bounds for trees and cubic graphs. For every cubic graph of order $ n, $ we prove that $ \gamma(G) \leq \frac{5n}{7} $ and show that this inequality is sharp. Also for every cubic graph of order $n$ with no subgraph $C_{5}$, we show that $\overline \gamma(G) \leq \frac{5n}{7}$. Moreover, it is proved that for every cubic graph $G$ of order $n$, except Petersen graph, $\overline{\gamma}(G) \le ...

Ars Mathematica Contemporanea, 2021

Complete vertex colorings have the property that any two color classes have at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers come from complete colorings, with some additional requirement. In this paper, we estimate these numbers in the Kneser graph K(n, k) for some values of n and k. We give the exact value of the achromatic number of K(n, 2).

Let G be a graph without isolated vertices and let α(G) be its stability number and τ (G) its covering number. The σ v -cover number of a graph, denoted by σ v (G), is the maximum natural number m such that every vertex of G belongs to a maximal independent set with at least m vertices. In the first part of this paper we prove that α

arXiv (Cornell University), 2021

A set D of vertices of a graph G is a dissociation set if each vertex of D has at most one neighbor in D. The dissociation number of G, diss(G), is the cardinality of a maximum dissociation set in a graph G. In this paper we study dissociation in the well-known class of Kneser graphs K n,k. In particular, we establish that the dissociation number of Kneser graphs Kn,2 equals max {n − 1, 6}. We show that for any k ≥ 2, there exists n0 ∈ N such that diss(K n,k) = α(K n,k) for any n ≥ n0. We consider the case k = 3 in more details and prove that n0 = 8 in this case. Then we improve a trivial upper bound 2α(K n,k) for the dissociation number of Kneser graphs K n,k by using Katona's cyclic arrangement of integers from {1,. .. , n}. Finally we investigate the odd graphs, that is, the Kneser graphs with n = 2k + 1. We prove that diss(K 2k+1,k) = 2k k .

Designs, Codes and Cryptography, 2012

We show that the q-Kneser graph qK 2k:k (the graph on the ksubspaces of a 2k-space over GF (q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q k + q k−1 for k = 3 and for k < q log q − q. We obtain detailed results on maximal cocliques for k = 3. d-space or just a [d]. We use projective terminology, so a point is a [1], a line a [2], a plane a [3], a solid a [4] and a hyperplane an [n−1] in V . Two [k]'s are adjacent in the Kneser graph if their intersection is [0]