On certain cycles in graphs

Journal of Combinatorial Theory, Series B, 1994

Let v = v(G) and e = e(G) denote the order and size of a simple graph G, respectively. Let G = {G i } i≥1 be a family of simple graphs of magnitude r > 1 and constant λ > 0, i.e. e(G i) = (λ + o(1))v(G i) r , i → ∞. For any such family G whose members are bipartite and of girth at least 2k + 2, and every integer t, 2 ≤ t ≤ k − 1, we construct a family G t of graphs of same magnitude r, of constant greater than λ, and all of whose members contain each of the cycles C 4 , C 6 ,. .. , C 2t , but none of the cycles C 2t+2 ,. .. , C 2k. We also prove that for every family of 2k-cycle free extremal graphs (i.e. graphs having the greatest size among all 2k-cycle free graphs of the same order), all but finitely many such graphs must be either non-bipartite or have girth at most 2k − 2. In particular, we show that the best known lower bound on the size of 2k-cycle free extremal graphs for k = 3, 5, namely (2 − k+1 k + o(1))v k+1 k , can be improved to ((k − 1) • k − k+1 k + o(1))v k+1 k .

Journal of Combinatorial Theory, Series B, 1981

In this paper we consider non-separating induced cycles in graphs. A basic result is that any 2-connected graph with at least six vertices and without such a cycle has at least four vertices of degree 2, and this is best possible. For any 3-connected graph G we prove that there exists a non-separating induced cycle C, such that all cycles in G-V(C) are contained in the same block of G-V(C). We apply our results in various directions. In particular, we obtain an extension of a conjecture of Hobbs (first proved by Jackson), and a new proof of Tutte's theorem on 3connected graphs. Moreover, we show that any graph with minimum degree at least 3 contains a subdivision of K a in which the three edges of a Hamiltonian path of the K4 are left undivided. This is an extension of a conjecture by Tort and implies an extension of a conjecture of Bollob/ts and Erd6s (first proved by Larson) on the existence of an odd cycle with at least one diagonal. Finally, we obtain a result on the existence of a vertex joined by edges to three vertices of a cycle in a graph. This implies an extremal result conjectured by Bollobas and Erdfs (first proved by Thomassen), as well as the conjecture of Toft that every 4-chromatic graph contains such a configuration.

Mathematical Problems in Engineering

Let G be a simple connected graph. Suppose Δ = Δ 1 , Δ 2 , … , Δ l an l -partition of V G . A partition representation of a vertex α w . r . t Δ is the l − vector d α , Δ 1 , d α , Δ 2 , … , d α , Δ l , denoted by r α | Δ . Any partition Δ is referred as resolving partition if ∀ α i ≠ α j ∈ V G such that r α i | Δ ≠ r α j | Δ . The smallest integer l is referred as the partition dimension pd G of G if the l -partition Δ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

SIAM Journal on Discrete Mathematics, 2018

Let L be a set of positive integers. We call a (directed) graph G an L-cycle graph if all cycle lengths in G belong to L. Let c(L, n) be the maximum number of cycles possible in an n-vertex L-cycle graph (we use c(L, n) for the number of cycles in directed graphs). In the undirected case we show that for any fixed set L, we have c(L, n) = Θ(n k/ ) where k is the largest element of L and 2 is the smallest even element of L (if L contains only odd elements, then c(L, n) = Θ(n) holds.) We also give a characterization of L-cycle graphs when L is a single element. In the directed case we prove that for any fixed set L we have c(L, n) = (1 + o( ))( n-1 k-1 ) k-1 , where k is the largest element of L. We determine the exact value of c({k}, n) for every k and characterize all graphs attaining this maximum.

Bulletin of the American Mathematical Society, 1971

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.