arXiv subjects
Publications and source records attributed to Paul Seymour.
We prove that for every path H, and every integer d, there is a polynomial f such that every graph G with chromatic number greater than f(t) either contains H as an induced subgraph, or contains as a subgraph the complete d-partite graph with parts of cardinality t. For t = 1 and general d this is a classical theorem of Gyárfás, and for d = 2 and general t this is a theorem of Bonamy et al.
In this paper we investigate the bipartite analogue of the strong Erdos-Hajnal property. We prove that for every forest $H$ and every $τ>0$ there exists $ε>0$, such that if $G$ has a bipartition $(A,B)$ and does not contain $H$ as an induced subgraph, and has at most $(1-τ)|A|\cdot|B|$ edges, then there is a stable set in $G$ that contains at least $ε|V_i|$ vertices of $V_i$, for $i=1,2$. No graphs $H$ except forests have this property.
The Gyárfás-Sumner conjecture says that for every tree $T$ and every integer $t\ge 1$, if $G$ is a graph with no clique of size $t$ and with sufficiently large chromatic number, then $G$ contains an induced subgraph isomorphic to $T$. This remains open, but we prove that under the same hypotheses, $G$ contains a subgraph $H$ isomorphic to $T$ that is ``path-induced''; that is, for some distinguished vertex~$r$, every path of $H$ with one end $r$ is an induced path of $G$.
We prove that every connected $P_5$-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected $P_5$-free graph $G$ with independence number at least three contains a three-vertex induced path with vertices $a \hbox{-} b \hbox{-} c$ in order, such that every neighbour of $c$ is also adjacent to one of $a,b$.
The Gyárfás-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an induced copy of $H$.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest $H$, if $G$ is $H$-free and does not contain $K_{t,t}$ as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest $H$, the bound on chromatic number can be taken to be polynomial in $t$. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree $H$ of radius two and every integer $d\ge 2$, if $G$ is $H$-free and does not contain as a subgraph the complete $d$-partite graph with parts of cardinality $t$, then its chromatic number is at most polynomial in $t$.
Let us say a graph is $s\mathcal{O}$-free, where $s\ge 1$ is an integer, if there do not exist $s$ cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when $s=2$, is not well understood. For instance, until now we did not know how to test whether a graph is $2\mathcal{O}$-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that $2\mathcal{O}$-free graphs have only a polynomial number of induced paths. In this paper we prove Le's conjecture; indeed, we will show that for all $s\ge 1$, there exists $c>0$ such that every $s\mathcal{O}$-free graph $G$ has at most $|G|^c$ induced paths. This provides a poly-time algorithm to test if a graph is $s\mathcal{O}$-free, for all fixed $s$. The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every $s\mathcal{O}$-free graph $G$ with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in $|G|$. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.
It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$ cliques suffice; and also prove that, even for graphs with no stable set of size four, we may need more than linearly many cliques. This settles two questions discussed at a recent conference in Lyon.
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 $ε>0$, there is a comparability graph $G$ with $n$ vertices, where $n$ is large, in which there is no pure pair $A,B$ with $|A|,|B|\ge εn$. He also proved that for all $c>0$ there exists $ε>0$ such that for every comparability graph $G$ with $n>1$ vertices, there is a pure pair $A,B$ with $|A|,|B|\ge εn^{1-c}$; and conjectured that the same holds for every perfect graph $G$. We prove this conjecture and strengthen it in several ways. In particular, we show that for all $c>0$, and all $\ell_1, \ell_2\ge 4c^{-1}+9$, there exists $ε>0$ such that, if $G$ is a graph with $n>1$ vertices and no hole of length exactly $\ell_1$ and no antihole of length exactly $\ell_2$, then there is a pure pair $A,B$ in $G$ with $|A|\ge εn$ and $|B|\ge εn^{1-c}$. This is further strengthened, replacing excluding a hole by excluding some long subdivision of a general graph.
A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $ω\ge 3$ has chromatic number at most $ω^{\log_2(ω)}$. The best previous result was an exponential upper bound $(5/27)3^ω$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that.
What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph F on n vertices is c-restricted if either F or its complement has maximum degree at most cn. Rodl proved that for every graph H, and every c>0, every H-free graph G has a linear-sized set of vertices inducing a c-restricted graph. We strengthen Rodl's result as follows: for every graph H, and all c>0, every H-free graph can be partitioned into a bounded number of subsets inducing c-restricted graphs.
Motivated by Hadwiger's conjecture, we prove that every $n$-vertex graph $G$ with no independent set of size three contains an $\lceil n/2\rceil$-vertex simple minor $H$ with $$0.98688 \cdot \binom{|V(H)|}{2} - o(n^2)$$ edges.
A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have $K_6$ minors, but minimum degree six is certainly not enough. For every $c>0$ there are arbitrarily large graphs with average degree at least $8-c$ and minimum degree at least six, with no $K_6$ minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $c>0$ there are arbitrarily large bipartite graphs with average degree at least $8-c$ and no $K_6$ minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a $K_6$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.
A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any $k$, if $G$ does not contain a $k$-multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially $χ$-bounded.
Say a graph $G$ is a {\em pentagraph} if every cycle has length at least five, and every induced cycle of odd length has length five. N. Robertson proposed the conjecture that the Petersen graph is the only pentagraph that is three-connected and internally 4-connected, but this was disproved by M. Plummer and X. Zha in 2014. Plummer and Zha conjectured that every 3-connected, internally 4-connected pentagraph is three-colourable. We prove this: indeed, we will prove that every pentagraph is three-colourable.
Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique $K$ such that for every vertex $v \in K$, the set of neighbours of $v$ outside of $K$ is a clique. In fact, we prove the existence of a simplicial clique in a more general class of graphs defined by forbidden induced subgraphs.
A double star is a tree with two internal vertices. It is known that the Gyárfás-Sumner conjecture holds for double stars, that is, for every double star $H$, there is a function $f$ such that if $G$ does not contain $H$ as an induced subgraph then $χ(G)\le f(ω(G))$ (where $χ, ω$ are the chromatic number and the clique number of $G$). Here we prove that $f$ can be chosen to be a polynomial.
Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.
The Gyarfas-Sumner conjecture says that for every forest $H$, there is a function $f$ such that if $G$ is $H$-free then $χ(G)\le f(ω(G))$ (where $χ, ω$ are the chromatic number and the clique number of $G$). Louis Esperet conjectured that, whenever such a statement holds, $f$ can be chosen to be a polynomial. The Gyarfas-Sumner conjecture is only known to be true for a modest set of forests $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when $H$ is a five-vertex path. Here we prove Esperet's conjecture when each component of $H$ is a star.