Browsing Waterloo Research by Author "Seymour, Paul"

Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs

Chudnovsky, Maria; Fox, Jacob; Scott, Alex; Seymour, Paul; Spirkl, Sophie (Wiley, 2020-11)

A graph is H-free if it has no induced subgraph isomorphic to H, and |G| denotes the number of vertices of G. A conjecture of Conlon, Sudakov and the second author asserts that: - For every graph H, there exists ∈ > 0 ...

Pure pairs. IV. Trees in bipartite graphs.

Scott, Alex; Seymour, Paul; Spirkl, Sophie (Elsevier, 2023-07)

In this paper we investigate the bipartite analogue of the strong Erd˝os-Hajnal property. We prove that for every forest H and every τ with 0 < τ ≤ 1, there exists ε > 0, such that if G has a bipartition (A,B) and does not ...

Pure Pairs. IX. Transversal Trees

Scott, Alex; Seymour, Paul; Spirkl, Sophie (Society for Industrial and Applied Mathematics, 2024)

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets ("blocks") of approximately equal size. An induced subgraph of G is "transversal" (with respect to this partition) if it has exactly one vertex in ...

Pure Pairs. V. Excluding Some Long Subdivision.

Scott, Alex; Seymour, Paul; Spirkl, Sophie (Springer, 2023-06-16)

A \pure pair" in a graph G is a pair A;B of disjoint subsets of V (G) such that A is complete or anticomplete to B. Jacob Fox showed that for all " > 0, there is a comparability graph G with n vertices, where n is large, ...

Pure pairs. VII. Homogeneous submatrices in 0/1-matrices with a forbidden submatrix

Scott, Alex; Seymour, Paul; Spirkl, Sophie (Elsevier, 2023-07)

For integer n>0, let f(n) be the number of rows of the largest all-0 or all-1 square submatrix of M, minimized over all n x n 0/1-matrices M. Thus f(n)=O(log n). But let us fix a matrix H, and define fH(n) to be the same, ...

Pure pairs. X. Tournaments and the strong Erdos-Hajnal property.

Chudnovsky, Maria; Scott, Alex; Seymour, Paul; Spirkl, Sophie (Elsevier ScienceDirect, 2024-01)

A pure pair in a tournament G is an ordered pair (A;B) of disjoint subsets of V (G) such that every vertex in B is adjacent from every vertex in A. Which tournaments H have the property that if G is a tournament not ...

Short Directed Cycles in Bipartite Digraphs

Seymour, Paul; Spirkl, Sophie (Springer Nature, 2020-08-01)

The Caccetta-Häggkvist conjecture implies that for every integer k ≥ 1, if G is a bipartite digraph, with n vertices in each part, and every vertex has out-degree more than n/(k+1), then G has a directed cycle of length ...

Strengthening Rodl's theorem

Chudnovsky, Maria; Scott, Alex; Seymour, Paul; Spirkl, Sophie (Elsevier, 2023-11)

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rödl showed in the 1980s that every H-free graph has large parts that are very sparse or very dense. More precisely, ...

Towards Erdős-Hajnal for Graphs with No 5-Hole

Chudnovsky, Maria; Fox, Jacob; Scott, Alex; Seymour, Paul; Spirkl, Sophie (Springer Nature, 2019-11-01)

The Erdős-Hajnal conjecture says that for every graph H there exists c > 0 such that max(α(G), w(G)) ≥ nc for every H-free graph G with n vertices, and this is still open when H = C5. Until now the best bound known on ...

Triangle-free graphs with no six-vertex induced path

Chudnovsky, Maria; Seymour, Paul; Spirkl, Sophie; Zhong, Mingxian (Elsevier, 2018-08)

The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced ...