Detecting a long odd hole

For each integer $t\ge 5$, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least $t$ and odd

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

Disjoint paths in 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 $k$ 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 for $k=2$ there is a polynomial-time algorithm when $G$ is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen. Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $G$ is semicomplete

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

Detecting a long odd hole

For each integer $t\ge 5$, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least $t$ and odd

Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyarfas' conjectures

Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chromatic number contains either a 5-hole or a hole of length more than $l$

The interrupted world: Surrealist disruption and altered escapes from reality

Following Breton’s writings on surreality, we outline how unexpected challenges to consumers’ assumptive worlds have the potential to alter how their escape from reality is experienced. We introduce the concept of ‘surrealist disruption’ to describe ontological discontinuities that disrupt the common-sense frameworks normally used by consumers and that impact upon their ability to suspend their disbeliefs and experience self-loss. To facilitate our theorization, we draw upon interviews with consumers about their changing experiences as viewers of the realist political TV drama House of Cards against a backdrop of disruptive real-world political events. Our analyses reveal that, when faced with a radically altered external environment, escape from reality changes from a restorative, playful experience to an uneasy, earnest one characterized by hysteretic angst, intersubjective sense-making and epistemological community-building. This reconceptualizes escapism as more emotionally multivalenced than previously considered in marketing theory and reveals consumers’ subject position to an aggregative social fabric beyond their control

Induced subgraphs of graphs with large chromatic number. III. Long holes

We prove a 1985 conjecture of Gy\'arf\'as that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$