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Paul Seymour

Publications and source records attributed to Paul Seymour.

At least 91 records · Page 5Linked to original sources
arXiv2019

Small families under subdivision

Let $H$ be a graph with maximum degree $d$, and let $d'\ge 0$. We show that for some $c>0$ depending on $H,d'$, and all integers $n\ge 0$, there are at most $c^n$ unlabelled simple $d$-connected $n$-vertex graphs with maximum degree at most $d'$ that do not contain $H$ as a subdivision. On the other hand, the number of unlabelled simple $(d-1)$-connected $n$-vertex graphs with minimum degree $d$ and maximum degree at most $d+1$ that do not contain $K_{d+1}$ as a subdivision is superexponential in $n$.

math.CO↗
arXiv2019

Induced subgraphs of graphs with large chromatic number. VI. Banana trees

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that every tree has this property; and in another earlier paper with M. Chudnovsky, we proved that every cycle has this property. Here we give a common generalization. Say a banana is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned. We also find some other multigraphs with the same property.

math.CO↗
arXiv2019

Short directed cycles in bipartite digraphs

The Caccetta-Häggkvist conjecture implies that for every integer $k\ge 1$, 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 at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $α, β> 0$ with $kα+β>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $β|B|$, and every vertex in $B$ has out-degree at least $α|A|$, then $G$ has a directed cycle of length at most $2k$. This implies the Caccetta-Häggkvist conjecture (set $β>0$ and very small), and again is best possible for infinitely many pairs $(α,β)$. We prove this for $k = 1,2$, and prove a weaker statement (that $α+β>2/(k+1)$ suffices) for $k=3,4$.

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arXiv2019

$H$-colouring $P_t$-free graphs in subexponential time

A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from $G$ to $H$ can be calculated in subexponential time $2^{O\left(\sqrt{tn\log(n)}\right)}$ for $n=|V(G)|$ in the class of $P_t$-free graphs $G$. As a corollary, we show that the number of 3-colourings of a $P_t$-free graph $G$ can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of $P_t$-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that $P_t$-free graphs have pathwidth that is linear in their maximum degree.

cs.DM↗
arXiv2019

Induced subgraphs of graphs with large chromatic number. VII. Gyárfás' complementation conjecture

A class of graphs is $χ$-bounded if there is a function $f$ such that $χ(G)\le f(ω(G))$ for every induced subgraph $G$ of every graph in the class, where $χ,ω$ denote the chromatic number and clique number of $G$ respectively. In 1987, Gyárfás conjectured that for every $c$, if $\mathcal{C}$ is a class of graphs such that $χ(G)\le ω(G)+c$ for every induced subgraph $G$ of every graph in the class, then the class of complements of members of $\mathcal{C}$ is $χ$-bounded. We prove this conjecture. Indeed, more generally, a class of graphs is $χ$-bounded if it has the property that no graph in the class has $c+1$ odd holes, pairwise disjoint and with no edges between them. The main tool is a lemma that if $C$ is a shortest odd hole in a graph, and $X$ is the set of vertices with at least five neighbours in $V(C)$, then there is a three-vertex set that dominates $X$.

math.CO↗
arXiv2019

Detecting an odd hole

A hole in a graph G is an induced cycle of length at least four; an antihole is a hole in the complement of G. In 2005, Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic showed that it is possible to test in polynomial time whether a graph contains an odd hole or antihole (and thus whether G is perfect). However, the complexity of testing for odd holes has remained open. Indeed, it seemed quite likely that testing for an odd hole was NP-complete: for instance, Bienstock showed that testing if a graph has an odd hole containing a given vertex is NP-complete. In this paper we resolve the question, by giving a polynomial-time algorithm to test whether a graph contains an odd hole. This also gives a new and considerably simpler polynomial-time algorithm that tests for perfection.

math.CO↗
arXiv2019

Near-domination in graphs

A vertex u of a graph t-dominates a vertex v if there are at most t vertices different from u,v that are adjacent to v and not to u; and a graph is t-dominating if for every pair of distinct vertices, one of them t-dominates the other. Our main result says that if a graph is t-dominating, then it is close (in an appropriate sense) to being 0-dominating. We also show that an analogous statement for digraphs is false; and discuss some connections with the Erdos-Hajnal conjecture.

math.CO↗
arXiv2018

Disjoint paths in unions of tournaments

Given $k$ pairs of vertices $(s_i,t_i)\;(1\le i\le k)$ of a digraph $G$, how can we test whether there exist vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but in an earlier paper we proved that for all fixed $k$, there is a polynomial-time algorithm to solve the problem if $G$ is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $V(G)$ is partitioned into a bounded number of sets each inducing a semicomplete digraph (and we are given the partition).

math.CO↗
arXiv2018

Induced subgraphs of graphs with large chromatic number. XII. Distant stars

The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain all double-ended brooms and two-legged caterpillars.

math.CO↗
arXiv2018

Induced subgraphs of graphs with large chromatic number. X. Holes of specific residue

A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. As an application, we prove two conjectures of Kalai and Meshulam from the 1990's connecting the chromatic number of a graph with the homology of its independence complex.

math.CO↗
arXiv2018

Caterpillars in Erdős-Hajnal

Let $T$ be a tree such that all its vertices of degree more than two lie on one path, that is, $T$ is a caterpillar subdivision. We prove that there exists $ε>0$ such that for every graph $G$ with $|V(G)|\ge 2$ not containing $T$ as an induced subgraph, either some vertex has at least $ε|V(G)|$ neighbours, or there are two disjoint sets of vertices $A,B$, both of cardinality at least $ε|V(G)|$, where there is no edge joining $A$ and $B$. A consequence is: for every caterpillar subdivision $T$, there exists $c>0$ such that for every graph $G$ containing neither of $T$ and its complement as an induced subgraph, $G$ has a clique or stable set with at least $|V(G)|^c$ vertices. This extends a theorem of Bousquet, Lagoutte and Thomassé [JCTB 2015], who proved the same when $T$ is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [Electron. J. Combin. 2018], who proved it when $T$ is a "hook".

math.CO↗
arXiv2018

Proof of the Kalai-Meshulam conjecture

Let $G$ be a graph, and let $f_G$ be the sum of $(-1)^{|A|}$, over all stable sets $A$. If $G$ is a cycle with length divisible by three, then $f_G= \pm 2$. Motivated by topological considerations, G. Kalai and R. Meshulam made the conjecture that,if no induced cycle of a graph $G$ has length divisible by three, then $|f_G|\le 1$. We prove this conjecture.

math.CO↗
arXiv2018

Induced subgraphs of graphs with large chromatic number. XI. Orientations

Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed out and many directed in). Aboulker et al subsequently conjectured that the answer is affirmative in both cases. We give affirmative answers to both questions.

math.CO↗
arXiv2018

Clustered Colouring in Minor-Closed Classes

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$, the clustered chromatic number of the class of $H$-minor-free graphs is tied to the tree-depth of $H$. In particular, if $H$ is connected with tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-colourable with monochromatic components of size at most $c(H)$. This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of $H$-minor-free graphs. If $t=3$ then we prove that 4 colours suffice, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.

math.CO↗
arXiv2018

Induced subgraphs of graphs with large chromatic number. XIII. New brooms

Gyárfás and Sumner independently conjectured that for every tree $T$, the class of graphs not containing $T$ as an induced subgraph is $χ$-bounded, that is, the chromatic numbers of graphs in this class are bounded above by a function of their clique numbers. This remains open for general trees $T$, but has been proved for some particular trees. For $k\ge 1$, let us say a broom of length $k$ is a tree obtained from a $k$-edge path with ends $a,b$ by adding some number of leaves adjacent to $b$, and we call $a$ its handle. A tree obtained from brooms of lengths $k_1,...,k_n$ by identifying their handles is a $(k_1,...,k_n)$-multibroom. Kierstead and Penrice proved that every $(1,...,1)$-multibroom $T$ satisfies the Gyárfás-Sumner conjecture, and Kierstead and Zhu proved the same for $(2,...,2)$-multibrooms. In this paper give a common generalization: we prove that every $(1,...,1,2,...,2)$-multibroom satisfies the Gyárfás-Sumner conjecture.

math.CO↗
arXiv2018

Large rainbow matchings in general graphs

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Barát, Gyárfás and Sárközy, we conjecture that if $n$ is odd then the same is true also in general graphs, and that if $n$ is even then $2n$ matchings of size $n$ suffice. We prove that any $3n-2$ matchings of size $n$ have a partial rainbow matching of size $n$.

math.CO↗
arXiv2018

Bad News for Chordal Partitions

Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question.

math.CO↗