Cycles of even length in graphs

Even cycles in graphs

✍ Joseph G. Conlon 📂 Article 📅 2004 🏛 John Wiley and Sons 🌐 English ⚖ 888 KB

## Abstract Let __G__ be a 3‐connected simple graph of minimum degree 4 on at least six vertices. The author proves the existence of an even cycle __C__ in __G__ such that __G‐V__(__C__) is connected and __G‐E__(__C__) is 2‐connected. The result is related to previous results of Jackson, and Thomas

Bipartite graphs with cycles of all even

Bipartite graphs with cycles of all even lengths

✍ Edward Schmeichel; John Mitchem 📂 Article 📅 1982 🏛 John Wiley and Sons 🌐 English ⚖ 428 KB 👁 1 views

## Abstract Let __G__ = (__X, Y, E__) be a bipartite graph with __X__ = __Y__ = __n__. Chvátal gave a condition on the vertex degrees of __X__ and __Y__ which implies that __G__ contains a Hamiltonian cycle. It is proved here that this condition also implies that __G__ contains cycles of every even

Cycles of even lengths modulo k

✍ Ajit A. Diwan 📂 Article 📅 2010 🏛 John Wiley and Sons 🌐 English ⚖ 84 KB

## Abstract Thomassen [J Graph Theory 7 (1983), 261–271] conjectured that for all positive integers __k__ and __m__, every graph of minimum degree at least __k__+1 contains a cycle of length congruent to 2__m__ modulo __k__. We prove that this is true for __k__⩾2 if the minimum degree is at least 2

Unavoidable cycle lengths in graphs

✍ Jacques Verstraete 📂 Article 📅 2005 🏛 John Wiley and Sons 🌐 English ⚖ 160 KB

## Abstract An old conjecture of Erdős states that there exists an absolute constant __c__ and a set __S__ of density zero such that every graph of average degree at least __c__ contains a cycle of length in __S__. In this paper, we prove this conjecture by showing that every graph of average degre

Lengths of cycles in halin graphs

✍ J. A. Bondy; L. Lovász 📂 Article 📅 1985 🏛 John Wiley and Sons 🌐 English ⚖ 436 KB 👁 1 views

A Halin graph is a plane graph H = T U C, where T is a plane tree with no vertex of degree t w o and at least one vertex of degree three or more, and C is a cycle connecting the endvertices of T in the cyclic order determined by the embedding of T We prove that such a graph on n vertices contains cy

Distribution of Cycle Lengths in Graphs

✍ Genghua Fan 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 148 KB

Bondy and Vince proved that every graph with minimum degree at least three contains two cycles whose lengths differ by one or two, which answers a question raised by Erdo ˝s. By a different approach, we show in this paper that if G is a graph with minimum degree d(G) \ 3k for any positive integer k,