Grzegorz Gancarzewicz
Czasopismo Techniczne, Nauki Podstawowe Zeszyt 2 NP (16) 2014, 2014, s. 37 - 43
https://doi.org/10.4467/2353737XCT.14.298.3386
The aim of this paper is to prove that if s > 1 and G is a graph of order n > 4s + 6 satisfying
2 > (4n - 4s - 3) / 3 ; then every matching of G lies on a cycle of length at least n-s and hence, in a path of length at least n - s + 1:
Grzegorz Gancarzewicz
Czasopismo Techniczne, Nauki Podstawowe Zeszyt 2 NP (16) 2014, 2014, s. 45 - 58
https://doi.org/10.4467/2353737XCT.14.299.3387
In this paper we prove that if G is a (k + 2)-connected graph on n > 3 vertices satisfying
P(n + k) :
dG(x; y) = 2 ) maxfd(x); d(y)g > n + k
2
for each pair of vertices x and y in G; then any path S G of length k is contained in a
hamiltonian cycle of G: