Showing papers on "Petersen graph published in 1982"
TL;DR: In this article, the authors discuss the family of implications C (m +, n − )→ C ( r +, s − ) for nonnegative integral values of m, n, r and s. The general question as to when such implications hold seems quite difficult.
Abstract: A graph G has property C ( m + , n − ) if for any choice of m + n points u 1 ,…, u m , v 1 ,…, v n in G there is a cycle in G which includes all of u 1 ,…, u m , but none of v 1 ,…, v n . We discuss the family of implications ‘ C ( m + , n − )→ C ( r + , s − )’ for various non-negative integral values of m , n , r and s . The general question as to when such implications hold seems quite difficult. We discuss some reductions of these problems and prove the implication C ( n + , 1 − )→ C ( n + 1 + , 0 − ) valid for n = 2, 3, and 4. The Petersen graph shows that this implication fails for n = 9.
12 citations
TL;DR: In this paper, it was shown that the minimum number of equivalence relations needed to cover a complete graph on n vertices with respect to a hamiltonian cycle tends to infinity with n.
Abstract: Let K n -C n denote the graph we obtain from the complete graph on n vertices by omitting the edges of a hamiltonian cycle. Following a conjecture of Duchet we prove that the minimum number of equivalence relations needed to cover this graph tends to infinity with n .
6 citations
TL;DR: It has been proved [1,2] that in a cubic 3-connected graph any 9 vertices lie on a common cycle.
Abstract: It has been proved [1,2] that in a cubic 3-connected graph any 9 vertices lie on a common cycle.
4 citations