Degree Sums, Connectivity and Dominating Cycles in Graphs

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.

Discrete Mathematics, 1990

Journal of Combinatorial Theory, Series B, 1989

Proceedings of the Edinburgh Mathematical Society, 1981

We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.

Discrete Mathematics, 1993

It was claimed by that if G is a connected graph of order at least 3 such that no bridge is incident to a vertex of degree 2 and no path contains three or more consecutive vertices of degree 2, then L'(G) is hamiltonian. By H.-J. , this was proved to be false. Necessary and sufficient conditions for corresponding counterexamples were also displayed. However, this characterization is not complete because some counterexamples were overlooked as is proved in this note. Furthermore, a counterexample to a theorem of about hamiltonian index of graphs with hamiltonian cyclic blocks was exhibited by . Here an alternative version of this theorem is presented.

Discrete Mathematics, 2007

In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, let f (k, ) be the maximum order of a graph G with independence number (G) , which has no k vertex-disjoint cycles. We prove that f (k, ) = 3k + 2 − 3 if 1 5 or 1 k 2, and f (k, ) 3k + 2 − 3 in general. We also prove the following results: (1) there exists a constant c (depending only on ) such that f (k, ) 3k + c , (2) there exists a constant t k (depending only on k) such that f (k, ) 2 + t k , and (3) there exists no absolute constant c such that f (k, ) c(k + ).

Information Processing Letters, 2006

Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u) + d(v) + δ(u, v) n + 1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u, v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.

Results in Mathematics, 1990

A set S of vertices of a graph G is dominating if each vertex z not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number,-y(G),

Canadian Journal of Mathematics, 1967

In this note, G will denote a finite undirected graph without multiple edges, and V = V(G) will denote its vertex set. The largest integer n for which G is n-vertex connected is the vertex-connectivity of G and will be denoted by λ = λ(G). One defines ζ to be the largest integer z not exceeding |V| such that for any set U ⊂ V with |U| = z, there is a cycle in G which contains U. The symbol i(U) will denote the component index of U. As a standard reference for this and other terminology, the authors recommend O. Ore (3).

Journal of Graph Theory, 1977

A variety of recent developments in hamiltonian theory are reviewed. In particular, several sufficient conditions for a graph to be hamiltonian, certain hamiltonian properties of line graphs, and various hamiltonian properties of powers of graphs are discussed. Furthermore, the concept of an n-distant hamiltonian graph is introduced and several theorems involving this special class of hamiltonian graphs are presented.