Cycles in hamiltonian graphs of prescribed maximum degree
نویسندگان
چکیده
Let G be a hamiltonian graph G of order n and maximum degree , and let C(G) denote the set of cycle lengths occurring in G. It is easy to see that |C(G)|¿ − 1. In this paper, we prove that if ¿n=2, then |C(G)|¿ (n + − 3)=2. We also show that for every ¿ 2 there is a graph G of order n¿ 2 such that |C(G)|= − 1, and the lower bound in case ¿n=2 is best possible. c © 2003 Elsevier Science B.V. All rights reserved. MSC: 05C45
منابع مشابه
On cycles in intersection graphs of rings
Let $R$ be a commutative ring with non-zero identity. We describe all $C_3$- and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. Also, we shall describe all complete, regular and $n$-claw-free intersection graphs. Finally, we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...
متن کاملA note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs
Let G be a balanced bipartite graph of order 2n ≥ 4, and let σ1,1(G) be the minimum degree sum of two non-adjacent vertices in different partite sets of G. In [On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163–165], Moon and Moser proved that if σ1,1(G) ≥ n+1, then G is hamiltonian. In this note, we show that if k is a positive integer, then the Moon-Moser condition also implies the...
متن کاملLong cycles in graphs with prescribed toughness and minimum degree
A cycle C of a graph G is a D~-cycle if every component of G V(C) has order less than 2. Using the notion of D~-cycles, a number of results are established concerning long cycles in graphs with prescribed toughness and minimum degree. Let G be a t-tough graph on n/> 3 vertices. If 6 > n/(t + 2) + 2 2 for some 2 ~< t + 1, then G contains a Dx-cycle. In particular, if 6>n/(t+ 1)--1, then G is ham...
متن کاملGraphs and colors : edge-colored graphs, edge-colorings and proper connections. (Graphes et couleurs : graphes arêtes-coloriés, coloration d'arêtes et connexité propre)
In this thesis, we study different problems in edge-colored graphs and edge-colored multigraphs, such as proper connection, strong edge colorings, and proper hamiltonian paths and cycles. Finally, we improve the known O(n) algorithm to decide the behavior of a graph under the biclique operator, by studying bicliques in graphs without false-twin vertices. In particular, • We first study the k-pr...
متن کاملOn two Hamilton cycle problems in random graphs
We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph Gn,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) lnn/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from Gn,p without destroy...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 266 شماره
صفحات -
تاریخ انتشار 2003