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 topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs $G$ and $H$, we consider the polyhedral complex $\text{Hom}(G,H)$ that parametrizes the directed graph homomorphisms $f: G \rightarrow H$. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals. We study examples of directed Hom complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of $\text{Hom}(G,T_n)$ for $T_n$ a tournament. Finally, we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified statements and proofs, other corrections and minor revisions incorporating comments from referee

Vertex cut of a graph and connectivity of its neighbourhood complex

We show that if a graph $G$ satisfies certain conditions then the connectivity of neighbourhood complex $\mathcal{N}(G)$ is strictly less than the vertex connectivity of $G$. As an application, we give a relation between the connectivity of the neighbourhood complex and the vertex connectivity for stiff chordal graphs, and for weakly triangulated graphs satisfying certain properties. Further, we prove that for a graph $G$ if there exists a vertex $v$ satisfying the property that for any $k$-subset $S$ of neighbours of $v$, there exists a vertex $v_S \neq v$ such that $S$ is subset of neighbours of $v_S$, then $\mathcal{N}(G-\{v\})$ is $(k-1)$-connected implies that $\mathcal{N}(G)$ is $(k-1)$-connected. As a consequence of this, we show that:(i) neighbourhood complexes of queen and king graphs are simply connected and (ii) if $G$ is a $(n+1)$-connected chordal graph which is not folded onto a clique of size $n+2$, then $\mathcal{N}(G)$ is $n$-connected.Comment: Comments are welcome