A note on chromatic number and induced odd cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyarfas and proved that if a graph G has no odd holes then chi(G) \u3c=( 2 omega(G)+2). Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K-4 nor odd holes then chi(G) \u3c= 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then chi(G) \u3c= 4 and chi(G) \u3c= 3 if G has radius at most 3, and for each vertex u of G, the set of vertices of the same distance to u induces abipartite subgraph. This answers some questions in [17]

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

Between 2- and 3-colorability

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n,\,1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$