arXiv subjects
Publications and source records attributed to Hossein Hajiabolhassan.
In this note, we prove that for any integer $n\geq 3$ the b-chromatic number of the Kneser graph $KG(m,n)$ is greater than or equal to $2{\lfloor {m\over 2} \rfloor \choose n}$. This gives an affirmative answer to a conjecture of [6].
In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in \cite{MR2279672} that $t^{\rm th}$ Mycielskian of the Kneser graph $KG(m,n)$ has the same circular chromatic number and chromatic number provided that $m+t$ is an even integer. We prove that if $m$ is large enough, then $χ(M^t(KG(m,n)))=χ_c(M^t(KG(m,n)))$ where $M^t$ is $t^{\rm th}$ Mycielskian. Also, we consider the generalized Kneser graph $KG(m,n,s)$ and show that there exists a threshold $m(n,s,t)$ such that $χ(M^t(KG(m,n,s)))=χ_c(M^t(KG(m,n,s)))$ for $m\geq m(n,s,t)$.
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.
In this paper we investigate some basic properties of fractional powers. In this regard, we show that for any rational number $1\leq {2r+1\over 2s+1}< og(G)$, $G^{2r+1\over 2s+1}\longrightarrow H$ if and only if $G\longrightarrow H^{-{2s+1\over 2r+1}}.$ Also, for two rational numbers ${2r+1\over 2s+1} < {2p+1\over 2q+1}$ and a non-bipartite graph $G$, we show that $G^{2r+1\over 2s+1} < G^{2p+1\over 2q+1}$. In the sequel, we introduce an equivalent definition for circular chromatic number of graphs in terms of fractional powers. We also present a sufficient condition for equality of chromatic number and circular chromatic number.
In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also consider Nešetřil's Pentagon problem. This problem is about the existence of high girth cubic graphs which are not homomorphic to the cycle of size five. Several problems which are closely related to Nešetřil's problem are introduced and their relations are presented.