Sparse Kneser graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words

On Colorings of Graph Powers

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 Ne\v{s}et\v{r}il'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 Ne\v{s}et\v{r}il's problem are introduced and their relations are presented

Colouring quadrangulations of projective spaces

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective space P^n has chromatic number n+2 or higher, unless G is bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996), 219-227]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser theorem

On semi-transitive orientability of Kneser graphs and their complements

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no edge between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an edge for all $0\leq i<j\leq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs include several important classes of graphs such as 3-colorable graphs, comparability graphs, and circle graphs, and they are precisely the class of word-representable graphs studied extensively in the literature. In this paper, we study semi-transitive orientability of the celebrated Kneser graph $K(n,k)$, which is the graph whose vertices correspond to the $k$-element subsets of a set of $n$ elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. We show that for $n\geq 15k-24$, $K(n,k)$ is not semi-transitive, while for $k\leq n\leq 2k+1$, $K(n,k)$ is semi-transitive. Also, we show computationally that a subgraph $S$ on 16 vertices and 36 edges of $K(8,3)$, and thus $K(8,3)$ itself on 56 vertices and 280 edges, is non-semi-transitive. $S$ and $K(8,3)$ are the first explicit examples of triangle-free non-semi-transitive graphs, whose existence was established via Erd\H{o}s' theorem by Halld\'{o}rsson et al. in 2011. Moreover, we show that the complement graph $\overline{K(n,k)}$ of $K(n,k)$ is semi-transitive if and only if $n\geq 2k$