[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-135384-en":3,"doc-seo-135384-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},135384,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Chi-boundedness of graphs containing no cycles with k chords","We prove that the family of graphs containing no cycle with exactly k chords is χ-bounded for all sufficiently large k, and also for k of the form ℓ(ℓ − 2) with integer ℓ ≥ 3. This result confirms, up to finitely many remaining values of k, a conjecture of Aboulker and Bousquet from 2015. The paper develops the structural framework needed to bound chromatic number in terms of clique number under these chord-restriction conditions.","arXiv :2208 . 14860v3 [math .CO] 30 Aug 2025  \nChi-boundedness of graphs containing no cycles with k chords  \nJoonkyung Lee∗ Shoham Letzter† Alexey Pokrovskiy‡  \nAbstract  \nWe prove that the family of graphs containing no cycle with exactly k-chords is χ-bounded, fork large enough or of form ℓ(ℓ − 2) with ℓ ≥ 3 an integer. This verifies (up to a finite number of values k) a conjecture of Aboulker and Bousquet (2015) .  \n1 Introduction  \nThe clique number of a graph G, denoted ω(G), is the size of its largest clique in G. The chromatic number of G, denoted χ (G), is the minimum number of colours in a proper vertex-colouring of G, which is a colouring of the vertices where adjacent vertices have distinct colours. It is easy to see that χ (G) ≥ ω (G) for every graph G, but the converse is far from the truth. Indeed, the chromatic number cannot be upper-bounded by a function of the clique number. This can be seen, for example, through a construction due to Mycielski [13] that provides a family of triangle-free graphs whose chromatic number is unbounded.  \nIn 1987, Gy´arf´as [8] proposed to study families of graphs for which the chromatic number can be upper-bounded in terms of the clique number. More precisely, Gy´arf´as called a family of graphs G χ-bounded if there is a function f such that χ (G) ≤ f (ω(G)) for every G ∈ G.  \nFor a graph F, denote by Forb(F) the family of graphs that contain no induced copy of F. This is a particularly interesting class of graphs for studying χ-boundedness, as it leads us to various examples and conjectures. Namely, one may ask: for which graphs F is Forb(F) χ-bounded? If F contains a cycle, Forb(F) is not χ-bounded; indeed, this follows from the existence of graphs with arbitrarily large chromatic number and girth (where the girth of a graph is the length of its shortest cycle), due to Erd˝os [6] . Gy´arf´as proved [8] that Forb(F) is χ-bounded when F is a path or a star. An intriguing conjecture regarding χ-boundedness, due to Gy´arf´as [7] and Sumner [16], asserts that  \n∗ Department of Mathematics, Yonsei University, Seoul, South Korea. E-mail: [joonkyunglee@yonsei.ac.kr](joonkyunglee@yonsei.ac.kr. Re)[. Re](joonkyunglee@yonsei.ac.kr. Re)search supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government MSITNRF-2022R1C1C1010300, Samsung STF Grant SSTF-BA2201-02, and IBS-R029-C4 .  \n†Department of Mathematics, University College London, London WC1E 6BT, UK. Email: [s.letzter](s.letzter@ucl.ac.uk)[@](s.letzter@ucl.ac.uk)[ucl.ac.uk](s.letzter@ucl.ac.uk). Research supported by the Royal Society.  \n‡Department of Mathematics, University College London, London WC1E 6BT, UK. Email: [a.pokrovskiy](a.pokrovskiy@ucl.ac.uk)[@](a.pokrovskiy@ucl.ac.uk)[ucl.ac.uk](a.pokrovskiy@ucl.ac.uk).  \nForb(F) is χ-bounded for every forest F. If true, this would solve the above question regarding graphs F for which Forb(F) is χ-bounded. The conjecture is known for some special cases, including all trees of radius 2 [9] and some trees of radius 3 [10], but is widely open in general.  \nFor a graph F, denote by Forb∗(F) the family of graphs that do not contain an induced copy of a subdivision of F, where a subdivision of F is a graph obtained by replacing edges of F by internally disjoint paths. Scott [15] proved the following weakening of the G´yarf´as–Sumner conjecture: Forb∗(F) is χ-bounded for every forest F. He also conjectured that Forb∗(F) is χ-bounded for every graph F, but this turned out to be false [4, 14] . At the best of our knowledge, there seems tobe no conjectured answer to the question that asks for which graphs F is Forb∗(F) χ-bounded.  \nThe fact that Forb∗(F) ⊆ Forb(F) for every graph F makes it natural to consider an ‘interpolation’between the two classes to ask an analogous question. More precisely, fix an edge subset E of F and consider the class F of graphs that contain no induced copy of a graph obtained from F by only subdividing the edges in E, while leavin","cbCailaCFgvC2iAA","https://ap.wps.com/l/cbCailaCFgvC2iAA","pdf",665999,1,38,"English","en",105,"# Introduction\n## Graphs, χ-boundedness, and clique number\n## Forbidden induced subgraphs and cycle/chord restrictions\n## Interpolation between induced subdivision restrictions\n## Main theorem and wheel-related discussion","[{\"question\":\"What does χ-bounded mean in this paper?\",\"answer\":\"A family of graphs is χ-bounded if there exists a function f such that for every graph G in the family, χ(G) ≤ f(ω(G)), where χ(G) is the chromatic number and ω(G) is the clique number.\"},{\"question\":\"What is the main result proved about cycles with k chords?\",\"answer\":\"For sufficiently large k, and for k = ℓ(ℓ − 2) with integer ℓ ≥ 3, the class of graphs with no cycle containing exactly k chords is χ-bounded, establishing the Aboulker–Bousquet conjecture up to finitely many k values.\"},{\"question\":\"How are chord-restricted cycles related to wheel-free questions?\",\"answer\":\"The paper notes that a k-wheel contains a cycle with exactly ℓ chords for all ℓ ≤ k − 2, motivating the study of related classes such as wheel-free graphs and their χ-boundedness (including known positive results and a known falsity for the strongest conjecture).\"}]","Chi-boundedness of graphs containing no cycles with k chords | 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does χ-bounded mean in this paper?","Question",{"text":76,"@type":77},"A family of graphs is χ-bounded if there exists a function f such that for every graph G in the family, χ(G) ≤ f(ω(G)), where χ(G) is the chromatic number and ω(G) is the clique number.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main result proved about cycles with k chords?",{"text":81,"@type":77},"For sufficiently large k, and for k = ℓ(ℓ − 2) with integer ℓ ≥ 3, the class of graphs with no cycle containing exactly k chords is χ-bounded, establishing the Aboulker–Bousquet conjecture up to finitely many k values.",{"name":83,"@type":74,"acceptedAnswer":84},"How are chord-restricted cycles related to wheel-free questions?",{"text":85,"@type":77},"The paper notes that a k-wheel contains a cycle with exactly ℓ chords for all ℓ ≤ k − 2, motivating the study of related classes such as wheel-free graphs and their χ-boundedness (including known positive results and a known falsity for the strongest conjecture).","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & 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