We extend the definition of the chromatic symmetric function X G to include graphs G with a vertex-weight function w : V (G) → N. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of X G .

We prove a conjecture of Liebenau, Pilipczuk, and the last two authors [8], that for every forest H there exists ε > 0, such that if G has n ≥ 2 vertices and does not contain H as an induced subgraph, then either• some vertex has degree at least εn; or • there are two disjoint sets A, B ⊆ V (G) with |A|, |B| ≥ εn, such that there are no edges between A, B.(It is known that no graphs H except forests have this property.) Consequently we prove that for every forest H, there exists c > 0 such that for every graph G containing neither H nor its complement as an induced subgraph, there is a clique or stable set of cardinality at least |V (G)| c .

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 √ tn log(n) 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.

We give a polynomial-time algorithm to test whether a graph contains an induced cycle with length more than three and odd.

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of the (k × k)-wall or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.1. For t ≥ 2, a t-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all k, t and ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every ∆ and subcubic subdivided caterpillar T , every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.