Excluding 4‐Wheels

(see Trotignon [121], Aboulker, Radovanović, Trotignon, and Vušković [4], Bousquet and Thomassé, [19], and Aboulker [1] for results and discussion related to both conjectures).…”

Section: Open Problemsmentioning

confidence: 99%

A survey of χ‐boundedness

Scott

Seymour

2020

Journal of Graph Theory

130

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“…If we do not require that the cycle is induced, then Aboulker [1] conjectures that the chromatic number is at most k (see Trotignon [109], Aboulker, Radovanović, Trotignon and Vušković [4], Bousquet and Thomassé, [15] and Aboulker [1] for results and discussion related to both conjectures).…”

Section: Cycles With Chordsmentioning

confidence: 99%

A survey of $χ$-boundedness

Scott,

Seymour

2018

Preprint

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If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now. ExamplesBefore we begin on any of the three main topics, let us give some useful graphs: different kinds of graphs, that are triangle-free (that is, they have clique number two) and have arbitrarily large chromatic number. The second example is just an existence proof rather than a construction, but the others are explicit constructions. Tutte's constructionThe first proof that triangle-free graphs of large chromatic number exist, is due to Tutte (writing as Blanche Descartes [34,35]), as follows. Let G 1 be a 1-vertex graph. Inductively, having defined G k , let G k have n vertices say; now take a set Y of k(n − 1) + 1 vertices, and for each n-subset X of Y take a copy of G k (disjoint from everything else), and join it to X by a matching. This makes G k+1 . It follows inductively that G k is not (k − 1)-colourable. Erdős' random graphErdős [38] proved that for all g, k ≥ 1, and all sufficiently large n, there is a graph G with n vertices in which every stable set has fewer than n/k vertices and every cycle has more than g edges. This can be shown as follows. Choose a function p = p(n) such that np → ∞ and (np) g = o(n). Now take a random graph G on 2n vertices, in which every pair of vertices is joined independently at random with probability p; then with probability tending to 1 as n → ∞, G has no stable set of cardinality at least n/k, and has at most n cycles of length at most g. Consequently, there is a set X of n vertices that intersects every cycle of length at most g, and by deleting X we obtain a graph with the desired properties. (Erdős actually made a more efficient argument deleting edges instead of vertices.) In particular, if we take g = 3, the graph we obtain is triangle-free and has chromatic number more than k.Lovász [77] and Nešetřil and Rödl [83] later found explicit constructions for such graphs. Mycielski's constructionMycielski [82] gave the following construction. Let G 2 be the two-vertex complete graph, and inductively, having defined G k , define G k+1 as follows. Let G k have vertex set {v 1 , . . . , v n }. Let G k+1 have 2n + 1 vertices a 1 , . . . , a n , b 1 , . . . , b n , c, wherethen a i a j , a i b j , b i a j are all edges of G k+1 ; andThen G k is triangle-free and has chromatic number k. Every triangle-free graph is an induced subgraph of G k for some k, so this is not a good source of graphs with forbidden induced subgraphs. Kneser graphsThe following grew from a problem of Kneser [68]. Let n, k be integers with n > 2k > 0, and let K(n, k) be the graph with vertex set the set of all k-subsets of {1, . . . , n}, in which two such sets are adjacent if they are disjoint. This graph has chromatic number n−2k+2, as was shown by Lov...

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Excluding 4‐Wheels

Abstract: A 4‐wheel is a graph formed by a cycle C and a vertex not in C that has at least four neighbors in C. We prove that a graph G that does not contain a 4‐wheel as a subgraph is 4‐colorable and we describe some structural properties of such a graph.

Cited by 2 publications

References 10 publications

A survey of χ‐boundedness

A survey of χ‐boundedness

A survey of $χ$-boundedness

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