A vertex cover with chorded 4-cycles

Let k be an integer with k ≥ 2 and G a graph with order n > 4k. We prove that if the minimum degree sum of any two nonadjacent vertices is at least n + k, then G contains a vertex cover with exactly k components such that k−1 of them are chorded 4-cycles. The degree condition is sharp in general.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

Spanning Trees with Few Leaves

In this paper, we prove that an m-connected graph G on n vertices has a spanning tree with at most k leaves (for k ≥ 2 and m ≥ 1) if every independent set of G with cardinality m + k contains at least one pair of vertices with degree sum at least n − k + 1. This is a common generalization of results due to Broersma and Tuinstra and to Win.     

Linear arboricity for graphs with multiple edges

Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ(G), the linear arboricity la(G) satisfies ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + 1)/2?. Here it is proved that if G is a loopless graph with maximum degree Δ(G) ≦ k and maximum edge multiplicity μ(G) ≦ k ? 2ⁿ⁺¹ + 1, where k ≧ 2^n?2, then la(G) ≦ k ? 2ⁿ. It is also conjectured that for any loopless graph G, ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + μ(G))/2?.     

On graphs with prescribed median I

The distance of a vertex u in a connected graph H is the sum of all the distances from u to the other vertices of H. The median M(H) of H is the subgraph of H induced by the vertices of minimum distance. For any graph G, let f(G) denote the minimum order of a connected graph H satisfying M(H) ? G. It is shown that if G has n vertices and minimum degree δ then f(G) ? 2n ? δ + 1. Graphs having both median and center prescribed are constructed. It is also shown that if the vertices of a K_r are removed from a graph H, then at most r components of the resulting graph contain median vertices of H.     

Complete subgraphs with large degree sums

Let t(n, k) denote the Turán number—the maximum number of edges in a graph on n vertices that does not contain a complete graph K_k+1. It is shown that if G is a graph on n vertices with n ≥ k²(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph K_k such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of K_k is not in general larger, and if the number of edges in G is at most t(n, k) – ? (for an appropriate ?), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.     

Proof of the Loebl–Komlós–Sós conjecture for large, dense graphs

The Loebl–Komlós–Sós conjecture states that for any integers k and n, if a graph G on n vertices has at least n/2 vertices of degree at least k, then G contains as subgraphs all trees on k+1 vertices. We prove this conjecture in the case when k is linear in n, and n is sufficiently large.     

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 a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

Cyclability of 3‐connected graphs

A subset S of vertices of a graph G is called cyclable in G if there is in G some cycle containing all the vertices of S. We denote by α(S, G) the number of vertices of a maximum independent set of G[S]. We prove that if G is a 3‐connected graph or order n and if S is a subset of vertices such that the degree sum of any four independent vertices of S is at least n + 2α(S, G) −2, then S is cyclable. This result implies several known results on cyclability or Hamiltonicity. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 191–203, 2000     

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.     

Spanning trees with many leaves

Let a maximal chain of vertices of degree 2 in a graph G consist of k > 0 vertices. We prove that G has a spanning tree with more than \fracv(G)2k + 4 \frac{{v(G)}}{{2k + 4}} leaves (where υ(G) is the number of vertices of the graph G). We present an infinite series of examples showing that the constant \frac12k + 4 \frac{1}{{2k + 4}} cannot be enlarged. Bibliography: 7 titles.     

Pentagons and cycle coverings

Let G be a graph of order n ≥ 5k + 2, where k is a positive integer. Suppose that the minimum degree of G is at least ?(n + k)/2?. We show that G contains k pentagons and a path such that they are vertex‐disjoint and cover all the vertices of G. Moreover, if n ≥ 5k + 7, then G contains k + 1 vertex‐disjoint cycles covering all the vertices of G such that k of them are pentagons. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 194–208, 2007     

A k-Tree Containing Specified Vertices

A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let ${S \subseteq V(G)}A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let S í V(G){S \subseteq V(G)} with κ(S) ≥ 1. Suppose that for every l > κ(S), there exists an integer t such that 1 ￡ t ￡ (k-1)l+2 - ?\fracl-1k ?{1 \le t \leq (k-1)l+2 - \lfloor \frac{l-1}{k} \rfloor} and the degree sum of any t independent vertices of S is at least n + tl − kl − 1. Then G has a k-tree containing S. We also show some new results on a spanning k-tree as corollaries of the above theorem.     

On a Spanning Tree with Specified Leaves

Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V(G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore’s theorem which guarantees the existence of a Hamilton path connecting any two vertices. Dedicated to Professor Hikoe Enomoto on his 60th birthday.     

An ore-type condition for the existence ofk-factors in graphs

Letk be a positive integer, and letG be a graph of ordern withn 4k – 5,kn even and minimum degree at leastk. We prove that if the degree sum of each pair of nonadjacent vertices is at leastn, thenG has ak-factor.     

Degree conditions for k‐ordered hamiltonian graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 ≤ k ≤ n/2, and deg(u) + deg(v) ≥ n + (3k − 9)/2 for every pair u, v of nonadjacent vertices of G, then G is k-ordered hamiltonian. Minimum degree conditions are also given for k-ordered hamiltonicity. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 199–210, 2003     

On the chromaticity of certain subgraphs of a q-tree

We show that a graph G on n ? q + 1 vertices (where q ? 2) has the chromatic polynomial P(G;λ) = λ(λ ? 1) … (λ ? q + 2) (λ ? q + 1)² (λ ? q)^n?q?1 if and only if G can be obtained from a q-tree Ton n vertices by deleting an edge contained in exactly q ? 1 triangles of T. Furthermore, we prove that these graphs are triangulated.     

Vertices of Degree 5 in a Contraction Critically 5-connected Graph

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is said to be contraction critically k-connected. We prove that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5. We also show that, for a graph G and an integer k greater than 4, there exists a contraction critically k-connected graph which has G as its induced subgraph.     

An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).