Making an $H$-Free Graph $k$-Colorable

Fox, Jacob; Zoe, Himwich,; Mani, Nitya

doi:10.48550/arxiv.2102.10220

Cited by 1 publication

(2 citation statements)

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“…This yields, for every fixed odd r, C r -free (n, d, λ)-graphs with d = Θ(n 2/r ) and λ = O( √ d). By (1.4), these graphs have surplus at most O(λn) = O((nd) (r+1)/(r+2) ), which shows that (1.3) would be optimal for all odd r. Regarding this problem, Zeng and Hou [23] showed that sp(m, C r ) ≥ m (r+1)/(r+3)+o (1) for all odd r, and very recently Fox, Himwich and Mani [14] improved the surplus to Ω r (m (r+5)/(r+7) ). We settle this problem completely by proving the following tight result.…”

Section: Cycles Of Odd Lengthmentioning

confidence: 96%

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New results for MaxCut in $H$-free graphs

Glock,

Janzer,

Sudakov

2021

Preprint

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The MaxCut problem asks for the size mc(G) of a largest cut in a graph G. It is well known that mc(G) ≥ m/2 for any m-edge graph G, and the difference mc(G) − m/2 is called the surplus of G. The study of the surplus of H-free graphs was initiated by Erdős and Lovász in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is Ω(m 4/5 ), and found constructions matching this bound. We prove several new results in this area.(i) We show that for every fixed odd r ≥ 3, any C r -free graph with m edges has surplus Ω r m r+1 r+2 . This is tight, as is shown by a construction of pseudorandom C r -free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when r is even.(ii) Generalizing the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs our bounds on the surplus are sharp.(iii) We prove that an n-vertex graph with few copies of K r and average degree d has surplus Ω r (d r−1 /n r−3 ), which is tight when d is close to n provided that a conjectured dense pseudorandom K r -free graph exists. This result is used to improve the best known lower bound (as a function of m) on the surplus of K r -free graphs.Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

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Section: Cycles Of Odd Lengthmentioning

confidence: 96%

“…The study of MaxCut in H-free graphs was initiated by Erdős and Lovász (see [12]) in the 70s, and has received significant attention since then (e.g. [2,3,5,6,14,20,21,23]). For a graph H, define sp(m, H) as the minimum surplus sp(G) = mc(G) − m/2 over all H-free graphs G with m edges.…”

Section: Introductionmentioning

confidence: 99%