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Stability in the Erdos--Gallai Theorem on cycles and paths | | Zoltán Füredi
; Alexandr Kostochka
; Jacques Verstraëte
; | | Date: |
19 Jul 2015 | | Abstract: | The ErdH{o}s-Gallai Theorem states that for $k geq 2$, every graph of
average degree more than $k - 2$ contains a $k$-vertex path. This result is a
consequence of a stronger result of Kopylov: if $tgeq 2$, $k=2t+1$, $n geq
frac{5t-3}{2}$, and $G$ is an $n$-vertex $2$-connected graph with at least
$h(n,k,t) = {k-t choose 2} + t(n -k+ t)$ edges, then $G$ contains a cycle of
length at least $k$ unless $G = H_{n,k,t} := K_n - E(K_{n - t})$. In this paper
we prove a stability version of the ErdH{o}s-Gallai Theorem: we show that for
all $n geq 3t > 3$, and $k in {2t+1,2t + 2}$, every $n$-vertex 2-connected
graph $G$ with $e(G) > h(n,k,t-1)$ either contains a cycle of length at least
$k$ or contains a set of $t$ vertices whose removal gives a star forest. In
particular, if $k = 2t + 1
eq 7$, we show $G subseteq H_{n,k,t}$. The lower
bound $e(G) > h(n,k,t-1)$ in these results is tight. | | Source: | arXiv, 1507.5338 | | Services: | Forum | Review | PDF | Favorites | |
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