Common graphs with arbitrary chromatic number

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F25%3A00384193" target="_blank" >RIV/68407700:21240/25:00384193 - isvavai.cz</a>

Alternative codes found

RIV/00216224:14330/25:00144082

Result on the web

<a href="http://hdl.handle.net/10467/125364" target="_blank" >http://hdl.handle.net/10467/125364</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1112/S0010437X24007681" target="_blank" >10.1112/S0010437X24007681</a>

Result language

angličtina

Original language name

Common graphs with arbitrary chromatic number

Original language description

Ramsey's theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erd & odblac;s conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of $K_k$ , and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko's conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common and, in fact, no 2-chromatic common graph unsettled for Sidorenko's conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known.We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami et al. [Non-three-colourable common graphs exist, Combin. Probab. Comput. 21 (2012), 734-742], and a problem listed by Conlon et al. [Recent developments in graph Ramsey theory, in Surveys in combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (Cambridge University Press, Cambridge, 2015), 49-118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger et al. [Multiplicities of subgraphs, Combinatorica 16 (1996), 123-131] whether there exists a common graph with chromatic number at least four.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GM23-06815M" target="_blank" >GM23-06815M: Extremal and probabilistic combinatorics</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Compositio Mathematica

ISSN

0010-437X

e-ISSN

1570-5846

Volume of the periodical

161

Issue of the periodical within the volume

3

Country of publishing house

GB - UNITED KINGDOM

Number of pages

41

Pages from-to

594-634

UT code for WoS article

001521860800001

EID of the result in the Scopus database

2-s2.0-105009968179