The chvátal-erdo˝s condition for cycles in triangle-free graphs
نویسندگان
چکیده
منابع مشابه
Dominating cycles in triangle-free graphs
A cycle C in a graph G is said to be dominating if E(G−C) = ∅. Enomoto et al. showed that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 2, then every longest cycle is dominating. But it is unknown whether the condition on the independence number is sharp. In this paper, we show that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 1, then G has a longest cycle which...
متن کاملCycles through specified vertices in triangle-free graphs
Let G be a triangle-free graph with δ(G) ≥ 2 and σ4(G) ≥ |V (G)|+ 2. Let S ⊂ V (G) consist of less than σ4/4 + 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ4 are best possible.
متن کاملLong Cycles in 2-Connected Triangle-Free Graphs
Dirac showed that a 2–connected graph of order n with minimum degree δ has circumference at least min{2δ, n}. We prove that a 2– connected, triangle-free graph G of order n with minimum degree δ either has circumference at least min{4δ−4, n}, or every longest cycle in G is dominating. This result is best possible in the sense that there exist bipartite graphs with minimum degree δ whose longest...
متن کاملCycles in triangle-free graphs of large chromatic number
More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of at least k2−ε different lengths as k →∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of 1 64 (1− ε)k 2 log k4 consecutive lengths, and a cycle of length at least 14 (1− ε)k 2 log k. As there ex...
متن کاملHamilton cycles in 1-tough triangle-free graphs
A graph G is called triangle-free if G has no induced K3 as a subgraph. We set 3 = min{3i=1 d(vi)|{v1; v2; v3} is an independent set of vertices in G}. In this paper, we show that if G is a 1-tough and triangle-free graph of order n with n6 3, then G is hamiltonian. c © 2002 Elsevier Science B.V. All rights reserved.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1996
ISSN: 0012-365X
DOI: 10.1016/0012-365x(96)80461-4