scholarly journals Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths

10.37236/6768 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Recent Result ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Large Chromatic Number ◽  
Vertex Colouring

We prove that for all integers $\kappa, s\ge 0$ there exists $c$ with the following property. Let $G$ be a graph with clique number at most $\kappa$ and chromatic number more than $c$. Then for every vertex-colouring (not necessarily optimal) of $G$, some induced subgraph of $G$ is an $s$-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016) who proved the same for graphs $G$ with $\kappa=2$ and girth at least five.

scholarly journals Induced subgraphs of graphs with large chromatic number. XI. Orientations

2019 ◽  
Vol 76 ◽  
pp. 53-61 ◽  
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Chromatic Number ◽  
Induced Subgraphs ◽  
Large Chromatic Number

scholarly journals Induced subgraphs of graphs with large chromatic number. VIII. Long odd holes

2020 ◽  
Vol 140 ◽  
pp. 84-97 ◽  
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour ◽  
Sophie Spirkl
Keyword(s):  
Chromatic Number ◽  
Induced Subgraphs ◽  
Large Chromatic Number

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

scholarly journals On Graphs Related to Comaximal Ideals of a Commutative Ring

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Tongsuo Wu ◽  
Meng Ye ◽  
Dancheng Lu ◽  
Houyi Yu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
The Core ◽  
Maximal Ideals ◽  
Graph Properties ◽  
Vertex Set

We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.

scholarly journals Induced subgraphs of graphs with large chromatic number. XII. Distant stars

2019 ◽  
Vol 92 (3) ◽  
pp. 237-254 ◽  
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Chromatic Number ◽  
Induced Subgraphs ◽  
Large Chromatic Number

scholarly journals Induced Subgraphs of Graphs with Large Chromatic Number. III. Long Holes

COMBINATORICA ◽  
2016 ◽  
Vol 37 (6) ◽  
pp. 1057-1072 ◽  
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Chromatic Number ◽  
Induced Subgraphs ◽  
Large Chromatic Number

scholarly journals Restricted Frame Graphs and a Conjecture of Scott

10.37236/4424 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jérémie Chalopin ◽  
Louis Esperet ◽  
Zhentao Li ◽  
Patrice Ossona de Mendez
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
Intersection Graphs

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once.  We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.

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