Degree Condition for Subdivisions of Unicyclic Graphs

If H is any graph of order n with k non-trivial components, each of which contains at most one cycle, then every graph of order at least n and minimum degree at least n − k contains a subdivision of H such that only edges contained in a cycle in H are subdivided.     

An Ore-Type Theorem on Hamiltonian Square Cycles

The kth power of a cycle C is the graph obtained from C by joining every pair of vertices with distance at most k on C. The second power of a cycle is called a square cycle. Pósa conjectured that every graph with minimum degree at least 2n/3 contains a hamiltonian square cycle. Later, Seymour proposed a more general conjecture that if G is a graph with minimum degree at least (kn)/(k + 1), then G contains the kth power of a hamiltonian cycle. Here we prove an Ore-type version of Pósa’s conjecture that if G is a graph in which deg(u) + deg(v) ≥ 4n/3 ? 1/3 for all non-adjacent vertices u and v, then for sufficiently large n, G contains a hamiltonian square cycle unless its minimum degree is exactly n/3 + 2 or n/3 + 5/3. A consequence of this result is an Ore-type analogue of a theorem of Aigner and Brandt.     

Spanning triangulations in graphs

We prove that every graph of sufficiently large order n and minimum degree at least 2n/3 contains a triangulation as a spanning subgraph. This is best possible: for all integers n, there are graphs of order n and minimum degree ?2n/3? ? 1 without a spanning triangulation. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.     

Tank-Ring Factors in Supereulerian Claw-Free Graphs

A graph G has a tank-ring factor F if F is a connected spanning subgraph with all vertices of degree 2 or 4 that consists of one cycle C and disjoint triangles attaching to exactly one vertex of C such that every component of G ? C contains exactly two vertices. In this paper, we show the following results. (1) Every supereulerian claw-free graph G with 1-hourglass property contains a tank-ring factor. (2) Every supereulerian claw-free graph with 2-hourglass property is Hamiltonian.     

Spanning Cyclic Subdivisions of Vertex-Disjoint Cycles and Chorded Cycles in Graphs

Let G be a graph on n ≥ 3 vertices and H be a subgraph of G such that each component of H is a cycle with at most one chord. In this paper we prove that if the minimum degree of G is at least n/2, then G contains a spanning subdivision of H such that only non-chord edges of H are subdivided. This gives a new generalization of the classical result of Dirac on the existence of Hamilton cycles in graphs.     

Spanning 3-ended trees in k-connected K 1,4-free graphs

A tree with at most m leaves is called an m-ended tree.Kyaw proved that every connected K1,4-free graph withσ4(G)n-1 contains a spanning 3-ended tree.In this paper we obtain a result for k-connected K1,4-free graphs with k 2.Let G be a k-connected K1,4-free graph of order n with k 2.Ifσk+3(G)n+2k-2,then G contains a spanning 3-ended tree.     

Note on Locating Pairs of Vertices on Hamiltonian Cycles

Given a fixed positive integer k ≥ 2, let G be a simple graph of order n ≥ 6k. It is proved that if the minimum degree of G is at least n/2 + 1, then for every pair of vertices x and y, there exists a Hamiltonian cycle such that the distance between x and y along that cycle is precisely k.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

Disjoint long cycles in a graph

We prove that if G is a graph of order at least 2k with k ? 9 and the minimum degree of G is at least k + 1, then G contains two vertex-disjoint cycles of order at least k. Moreover, the condition on the minimum degree is sharp.     

Topological Subgraphs in Graphs of Large Girth

H of maximum degree , there is an integer g(H) such that every finite graph of minimum degree n and girth at least g(H) contains a subdivision of H. This had been conjectured for in [8]. We prove also that every finite 2n-connected graph of sufficiently large girth is n-linked, and this is best possible for all . Received: February 26, 1997     

Edge-dominating cycles in graphs

A set S of vertices in a graph G is said to be an edge-dominating set if every edge in G is incident with a vertex in S. A cycle in G is said to be a dominating cycle if its vertex set is an edge-dominating set. Nash-Williams [Edge-disjoint hamiltonian circuits in graphs with vertices of large valency, Studies in Pure Mathematics, Academic Press, London, 1971, pp. 157-183] has proved that every longest cycle in a 2-connected graph of order n and minimum degree at least is a dominating cycle. In this paper, we prove that for a prescribed positive integer k, under the same minimum degree condition, if n is sufficiently large and if we take k disjoint cycles so that they contain as many vertices as possible, then these cycles form an edge-dominating set. Nash-Williams’ Theorem corresponds to the case of k=1 of this result.     

The Erd s-Sós conjecture for graphs of girth 5

We prove that every graph of girth at least 5 with minimum degree δ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erd s-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.     

Edge disjoint cycles in graphs

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ? 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ? 2δ², then for any 3 ? l₁ ? l₂ ? ? ? l_k ? n, 1 = k = [(δ - 1)/2], such graph has K edge disjoint cycles with lengths l₁, l₂…l_k, respectively. In particular, when l₁ = l₂ = ? = l_k = n and k = [(δ - 1)/2], the graph contains [(δ - 1)/2] edge disjoint hamiltonian cycles.     

A Note on Large Rainbow Matchings in Edge-coloured Graphs

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on n vertices with minimum colour degree at least k contains a rainbow matching of size at least k, provided ${n\geq \frac{17}{4}k^2}$ . In this paper, we show that n ≥ 4k ? 4 is sufficient for k ≥ 4.     

On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G - V(S).For an integer k ≥ 4,...     

Four gravity results

The gravity of a graph H in a given family of graphs H is the greatest integer n with the property that for every integer m, there exists a supergraph G∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n vertices of degree ?m in G. Madaras and Škrekovski introduced this concept and showed that the gravity of the path P_k on k?2 vertices in the family of planar graphs of minimum degree 2 is k-2 for each k≠5,7,8,9. They conjectured that for each of the four excluded cases the gravity is k-3. In this paper we show that this holds.     

The longest cycle of a graph with a large minimal degree

We show that every graph G on n vertices with minimal degree at least n/k contains a cycle of length at least [n/(k ? 1)]. This verifies a conjecture of Katchalski. When k = 2 our result reduces to the classical theorem of Dirac that asserts that if all degrees are at least 1/2n then G is Hamiltonian.     

Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P₄, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K₁+(K_1,k+1∪(n−k−3)K₁).     

The minimum spanning strong subdigraph problem is fixed parameter tractable

A digraph D is strong if it contains a directed path from x to y for every choice of vertices x,y in D. We consider the problem (MSSS) of finding the minimum number of arcs in a spanning strong subdigraph of a strong digraph. It is easy to see that every strong digraph D on n vertices contains a spanning strong subdigraph on at most 2n−2 arcs. By reformulating the MSSS problem into the equivalent problem of finding the largest positive integer k≤n−2 so that D contains a spanning strong subdigraph with at most 2n−2−k arcs, we obtain a problem which we prove is fixed parameter tractable. Namely, we prove that there exists an O(f(k)n^c) algorithm for deciding whether a given strong digraph D on n vertices contains a spanning strong subdigraph with at most 2n−2−k arcs.We furthermore prove that if k≥1 and D has no cut vertex then it has a kernel of order at most (2k−1)². We finally discuss related problems and conjectures.