We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any  triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Alon and Mohar (2002) posed the following problem: among all graphs G of maximum degree at most d and girth at least g, what is the largest possible value of χ(G t ), the chromatic number of the t th power of G? For t ≥ 3, we provide several upper and lower bounds concerning this problem, all of which are sharp up to a constant factor as d → ∞. The upper bounds rely in part on the probabilistic method, while the lower bounds are various direct constructions whose building blocks are incidence structures.

Any triangle-free graph on n vertices with minimum degree at least d contains a bipartite induced subgraph of minimum degree at least d 2 /(2n). This is sharp up to a logarithmic factor in n. We also provide a related extremal result for the fractional chromatic number.

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.

Given a graph G, the strong clique number ω 2 (G) of G is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in G. We study the strong clique number of graphs missing some set of cycle lengths. For a graph G of large enough maximum degree ∆, we show among other results the following:These bounds are attained by natural extremal examples. Our work extends and improves upon previous work of Faudree, Gyárfás, Schelp and Tuza (1990), Mahdian (2000) and Faron and Postle (2019). We are motivated by the corresponding problems for the strong chromatic index.

A code X is k-circular if any concatenation of at most k words from X , when read on a circle, admits exactly one partition into words from X. It is circular if it is k-circular for every integer k. While it is not a priori clear from the definition, there exists, for every pair (n,), an integer k such that every k-circular-letter code over an alphabet of cardinality n is circular, and we determine the least such integer k for all values of n and. The k-circular codes may represent an important evolutionary step between the circular codes, such as the comma-free codes, and the genetic code.

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