Combinatorica最新文献

We prove that, for every graph F with at least one edge, there is a constant (c_F) such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F-free induced subgraph has chromatic number at most (c_F). This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every (rgeqslant 3) the class of (K_r)-free graphs has a very strong vertex Ramsey-type property, giving a vast generalisation of a result of Folkman from 1970. We also prove related results for tournaments, hypergraphs and infinite families of graphs, and show an analogous statement for graphs where clique number is replaced by odd girth.

We give an algorithm with complexity (O((R+1)^{4k^2} k^3 n)) for the integer multiflow problem on instances (G, H, r, c) with G an acyclic planar digraph and (r+c) Eulerian. Here, (n = |V(G)|), (k = |E(H)|) and R is the maximum request (max _{h in E(H)} r(h)). When k is fixed, this gives a polynomial-time algorithm for the arc-disjoint paths problem under the same hypothesis.Kindly check and confirm the edit made in the title.ConfirmedJournal instruction requires a city and country for affiliations; however, these are missing in affiliation [1]. Please verify if the provided city is correct and amend if necessary.Since the submission, my affiliation has changed. It should now be:Laboratoire d'Informatique & Systèmes, Aix-Marseille Université, CNRS UMR 7020, Marseille, France

Let (A subset {mathbb {Z}}^d) be a finite set. It is known that NA has a particular size ((vert NAvert = P_A(N)) for some (P_A(X) in {mathbb {Q}}[X])) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when (d=1), when the convex hull of A is a simplex or when (vert Avert = d+2) Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.

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 (omega ge 3) has chromatic number at most (omega ^{log _2(omega )}). The best previous result was an exponential upper bound ((5/27)3^{omega }), due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-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.

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function (f:mathbb {N}rightarrow mathbb {N}cup {infty }) with (f(1)=1) and (f(n)geqslant left( {begin{array}{c}3n+1 3end{array}}right) ), we construct a hereditary class of graphs ({mathcal {G}}) such that the maximum chromatic number of a graph in ({mathcal {G}}) with clique number n is equal to f(n) for every (nin mathbb {N}). In particular, we prove that there exist hereditary classes of graphs that are (chi )-bounded but not polynomially (chi )-bounded.

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.