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

Fullerene graphs of small diameter

A fullerene graph is a cubic bridgeless plane graph with only pentagonal and hexagonal faces. We exhibit an infinite family of fullerene graphs of diameter $\sqrt{4n/3}$, where $n$ is the number of vertices. This disproves a conjecture of Andova and \v{S}krekovski [MATCH Commun. Math. Comput. Chem. 70 (2013) 205-220], who conjectured that every fullerene graph on $n$ vertices has diameter at least $\lfloor \sqrt{5n/3}\rfloor-1$

Replication in critical graphs and the persistence of monomial ideals

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose replication produces a (k+1)-critical graph. (The replication of a set W of vertices of a graph is the operation that adds a copy of each vertex w in W, one at a time, and connects it to w and all its neighbours.) We disprove the conjecture by providing an infinite family of counterexamples. Furthermore, the smallest member of the family answers a question of Herzog and Hibi concerning the depth functions of square-free monomial ideals in polynomial rings, and a related question on the persistence property of such ideals

Edge-critical subgraphs of Schrijver graphs II: The general case

We give a simple combinatorial description of an $(n-2k+2)$-chromatic edge-critical subgraph of the Schrijver graph $\mathrm{SG}(n,k)$, itself an induced vertex-critical subgraph of the Kneser graph $\mathrm{KG}(n,k)$. This extends the main result of [J. Combin. Theory Ser. B 144 (2020) 191--196] to all values of $k$, and sharpens the classical results of Lov\'asz and Schrijver from the 1970s

Fan's lemma via bistellar moves

Pachner proved that all closed combinatorially equivalent combinatorial manifolds can be transformed into each other by a finite sequence of bistellar moves. We prove an analogue of Pachner's theorem for combinatorial manifolds with a free Z2-action, and use it to give a combinatorial proof of Fan's lemma about labellings of centrally symmetric triangulations of spheres. Similarly to other combinatorial proofs, we must assume an additional property of the triangulation for the proof to work. However, unlike the other combinatorial proofs, no such assumption is needed for dimensions at most 3

Every plane graph of maximum degree 8 has an edge-face 9-colouring

An edge-face colouring of a plane graph with edge set $E$ and face set $F$ is a colouring of the elements of $E \cup F$ such that adjacent or incident elements receive different colours. Borodin proved that every plane graph of maximum degree $\Delta\ge10$ can be edge-face coloured with $\Delta+1$ colours. Borodin's bound was recently extended to the case where $\Delta=9$. In this paper, we extend it to the case $\Delta=8$.Comment: 29 pages, 1 figure; v2 corrects a contraction error in v1; to appear in SIDM

Connected τ -critical hypergraphs of minimal size

A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -critical if and only if it is $χ$ -critical 3-chromatic. Finally, we deduce some new results on $χ$ -vertex-critical graphs with connected complements

On the contractibility of random Vietoris-Rips complexes

We show that the Vietoris-Rips complex $\mathcal R(n,r)$ built over $n$ points sampled at random from a uniformly positive probability measure on a convex body $K\subseteq \mathbb R^d$ is a.a.s. contractible when $r \geq c \left(\frac{\ln n}{n}\right)^{1/d}$ for a certain constant that depends on $K$ and the probability measure used. This answers a question of Kahle [Discrete Comput. Geom. 45 (2011), 553-573]. We also extend the proof to show that if $K$ is a compact, smooth $d$-manifold with boundary - but not necessarily convex - then $\mathcal R(n,r)$ is a.a.s. homotopy equivalent to $K$ when $c_1 \left(\frac{\ln n}{n}\right)^{1/d} \leq r \leq c_2$ for constants $c_1=c_1(K), c_2=c_2(K)$. Our proofs expose a connection with the game of cops and robbers.Comment: 15 page