The Structure of Typical Eye-Free Graphs and a Turán-Type Result for Two Weighted Colours
The (a,b)-eye is the graph I a,b = K a+b \ K b obtained by deleting the edges of a clique of size b from a clique of size a+b. We show that for any a,b ≥ 2 and p ∈ (0,1), if we condition the random graph G ~ G(n,p) on having no induced copy of I a,b , then with high probability G is close to an a-pa...
Saved in:
| Published in | Combinatorics, probability & computing Vol. 26; no. 6; pp. 886 - 910 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.11.2017
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Summary: | The (a,b)-eye is the graph I
a,b
= K
a+b
\ K
b
obtained by deleting the edges of a clique of size b from a clique of size a+b. We show that for any a,b ≥ 2 and p ∈ (0,1), if we condition the random graph G ~ G(n,p) on having no induced copy of I
a,b
, then with high probability G is close to an a-partite graph or the complement of a (b−1)-partite graph. Our proof uses the recently developed theory of hypergraph containers, and a stability result for an extremal problem with two weighted colours. We also apply the stability method to obtain an exact Turán-type result for this extremal problem. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0963-5483 1469-2163 |
| DOI: | 10.1017/S0963548317000293 |