On certain cycles in graphs

Journal of Graph Theory, 2005

An old conjecture of Erdo ˝s 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 degree at least ten contains a cycle of length in a prescribed set S satisfying jS \ f1; 2; . . . ; ngj ¼ O(n 0:99 ).

COMBINATORICA, 2004

A set S of integers is called a cycle set on {1, 2,. .. , n} if there exists a graph G on n vertices such that the set of lengths of cycles in G is S. Erdős conjectured that the number of cycle sets on {1, 2,. .. , n} is o(2 n). In this paper, we verify this conjecture by proving that there exists an absolute constant c ≥ 0.1 such that the number of cycle sets on {1, 2,. .. , n} is o(2 n−n c).

Journal of Combinatorial Theory, Series B, 2006

Improving a result of Erdős, Gyárfás and Pyber for large n we show that for every integer r 2 there exists a constant n 0 = n 0 (r) such that if n n 0 and the edges of the complete graph K n are colored with r colors then the vertex set of K n can be partitioned into at most 100r log r vertex disjoint monochromatic cycles.

Information Processing Letters, 2012

A graph G = (V , E) of order n is called arbitrarily partitionable, or AP for short, if given any sequence of positive integers n 1 , . . . ,n k summing up to n, we can always partition additionally G is minimal with respect to this property, i.e. it contains no AP spanning subgraph, we call it a minimal AP-graph. It has been conjectured that such graphs are sparse, i.e., there exists an absolute constant C such that |E| Cn for each of them. We construct a family of minimal AP-graphs which prove that C 1 + 1 30 (if such C exists).

Discrete Mathematics, 2004

For c ¿ 2 and k 6 min{c; 3}, guaranteed upper bounds on the length of a shortest cycle through k prescribed vertices of a c-connected graph are proved. Analogous results on planar graphs are presented, too.

Combinatorics, Probability and Computing, 2000

A question recently posed by Häggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

1999

We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power sums.

Discrete Mathematics, 2005

Let n, h be integers with n ≥ 6 and h ≥ 7. We prove that if G is a graph of order n with σ 2 (G) ≥ h, then G contains two disjoint cycles C 1 and C 2 such that |V (C 1)| + |V (C 2)| ≥ min{h, n}.

Combinatorics, Probability and Computing, 2002

Häggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k2+19k+10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k2 cannot be decreased and, in this sense, this result is best possible.

International Journal of Mathematics and Mathematical Sciences, 1994

ABSTP,.ACT. Let q pn be a power of an odd prime p. We show that the t(q) vertices of every graph G can be partitioned into t(q) classes V(G)= 13 V such that the number of t=l edges in any induced subgraph < Vi> is divisible by q, where t(q)<_ (q-l) -(2(q-1)-1)4 if q 2n, then t(q) 2q-1.