Degree sets for homogeneously traceable non-Hamiltonian graphs
Spanning caterpillars with bounded diameter
A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most (at most n).
Degree sets and graph factorizations
A note on the Song-Zhang theorem for Hamiltonian graphs
An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. In 1994, Song and Zhang proved that if for each independent set S of cardinality k+1, one of the following condition holds: (i) there exist u ≠ v ∈ S such that d(u) + d(v) ≥ n or |N(u) ∩ N(v)| ≥ α (G); (ii) for any distinct u and v in S, |N(u) ∪ N(v)| ≥ n - max{d(x): x ∈ S}, then G is Hamiltonian. We prove that if for each essential...
Constructive upper bounds for cycle-saturated graphs of minimum size.
Weak Saturation Numbers for Sparse Graphs
For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order...
On the Non-(p−1)-Partite Kp-Free Graphs
We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m?
2-factors in claw-free graphs
We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced .) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ (n-2)/3, G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ (n-24)/3. We also show that this result is sharp in the sense that if we lower δ(G), we cannot obtain the full range of values for k.
Forbidden triples implying Hamiltonicity: for all graphs
In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with for some i = 1, 2, or 3, such that all G₁G₂G₃-free graphs contain a hamiltonian cycle. In [6], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁, G₂, G₃, none of which is a , s ≥ 3 such that G₁, G₂, G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be given of all triples G₁, G₂, G₃ with none being , such that all G₁G₂G₃-free...
Potential forbidden triples implying hamiltonicity: for sufficiently large graphs
In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a , s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being , such that all G₁G₂G₃-free graphs are...
The structure and existence of 2-factors in iterated line graphs
We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where contains a 2-factor.
The min-max supergraph
The Chvátal-Erdős condition and 2-factors with a specified number of components
Let G be a 2-connected graph of order n satisfying α(G) = a ≤ κ(G), where α(G) and κ(G) are the independence number and the connectivity of G, respectively, and let r(m,n) denote the Ramsey number. The well-known Chvátal-Erdös Theorem states that G has a hamiltonian cycle. In this paper, we extend this theorem, and prove that G has a 2-factor with a specified number of components if n is sufficiently large. More precisely, we prove that (1) if n ≥ k·r(a+4, a+1), then G has a 2-factor with k components,...
Chvátal-Erdös type theorems
The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-k)/(k+1),...
Saturation Spectrum of Paths and Stars
A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from G̅ to G results in a copy of H. The minimum size of an H-saturated graph on n vertices is denoted sat(n,H), while the maximum size is the well studied extremal number, ex(n,H). The saturation spectrum for a graph H is the set of sizes of H saturated graphs between sat(n,H) and ex(n,H). In this paper we completely determine the saturation spectrum of stars and we show the saturation spectrum of paths is continuous...
On -ordered bipartite graphs.
Saturation numbers of books.
Linear forests and ordered cycles
A collection (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian.
Saturation numbers for trees.
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