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.
Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f (k)n O(1) and space f (k) log(n) (for some computable function f ). We give a wide variety of XNLP-complete problems, such as LIST COLORING and PRECOLORING EXTENSION with pathwidth as parameter, SCHEDULING OF JOBS WITH PRECEDENCE CONSTRAINTS, with both number of machines and partial order width as parameter, BANDWIDTH and variants of WEIGHTED CNF-SATISFIABILITY and reconfiguration problems. In particular, this implies that all these problems are W[t]-hard for all t. This also answers a long standing question on the parameterized complexity of the BANDWIDTH problem.
An independent transversal of a graph G with a vertex partition is an independent set of G intersecting each block of in a single vertex. Wanless and Wood proved that if each block of has size at least t and the average degree of vertices in each block is at most t 4 ∕ , then an independent transversal of exists. We present a construction showing that this result is optimal: for any ε > 0 and sufficiently large t, there is a forest with a vertex partition into parts of size at least t such that the average degree of vertices in each block is, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science.We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.M. Bonamy and N. Bousquet are supported by ANR Projects DISTANCIA (ANR-17-CE40-0015) and GrR (ANR-18-CE40-0032). L. Esperet and F. Pirot are supported by ANR Projects GATO (ANR-16-CE40-0009-01) and GrR (ANR-18-CE40-0032). C.-H. Liu is partially supported by NSF under Grant No. DMS-1929851 and DMS-1954054.
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