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.     

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 Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces

 We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path P _k , a path on k vertices, such that each of its k vertices has degree ≤5/3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k∈ {5,8,11,14}, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k≥ 3, k∉ {5,8,11,14} and m≥ 3, we present a graph in which every k-cycle contains a vertex of degree at least m. Received: June 29, 1998 Final version received: April 11, 2000     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

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.     

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.     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

SCORE LISTS IN （h, k）-BIPARTITE HYPERTOURNAMENTS

Given non-negative integers m,n,h and k with m ≥ h ＞ 1 and n ≥ k ＞ 1, an (h, k)-bipartite hypertournament on m n vertices is a triple (U, V, A), where U and V are two sets of vertices with |U| = m and |V| = n, and A is a set of (h k)-tuples of vertices,called arcs, with at most h vertices from U and at most k vertices from V, such that for any h k subsets U1 ∪ V1 of U ∪ V, A contains exactly one of the (h k)! (h k)-tuples whose entries belong to U1 ∪ V1. Necessary and sufficient conditions for a pair of non-decreasing sequences of non-negative integers to be the losing score lists or score lists of some(h, k)-bipartite hypertournament are obtained.     

Spanning k-Trees of n-Connected Graphs

A tree is called a k-tree if the maximum degree is at most k. We prove the following theorem, by which a closure concept for spanning k-trees of n-connected graphs can be defined. Let k ≥ 2 and n ≥ 1 be integers, and let u and v be a pair of nonadjacent vertices of an n-connected graph G such that deg _G (u) + deg _G (v) ≥ |G| − 1 − (k − 2)n, where |G| denotes the order of G. Then G has a spanning k-tree if and only if G + uv has a spanning k-tree.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

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.     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

Topological minors in bipartite graphs

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K _(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.     

Large Vertex-Disjoint Cycles in a Bipartite Graph

Let s≥2 and k be two positive integers. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n≥s k and the minimum degree at least (s−1)k+1. When s=2 and n >2k, it is proved in [5] that G contains k vertex-disjoint cycles. In this paper, we show that if s≥3, then G contains k vertex-disjoint cycles of length at least 2s. Received: March 2, 1998 Revised: October 26, 1998     

Graphs with unavoidable subgraphs with large degrees

Let ??(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ?? (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn², α > (k - 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ?n²/4? + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn², α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

Covering a graph with cycles

Let k and n be two integers such that k ≥ 0 and n ≥ 3(k + 1). Let G be a graph of order n with minimum degree at least ?(n + k)/2?. Then G contains k + 1 independent cycles covering all the vertices of G such that k of them are triangles. © 1995, John Wiley & Sons, Inc.     

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any v ? V(G){v \in V(G)} , let N(v)={u ? V(G): uv ? E(G)}{N(v)=\{u \in V(G): uv \in E(G)\}} , NC(G) = min{|N(u) èN(v)|: u, v ? V(G){NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)} and uv \not ? E(G)}{uv \not \in E(G)\}} , and NC₂(G) = min{|N(u) èN(v)|: u, v ? V(G){NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any u, v ? V(G){u, v \in V(G)} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.     

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any , let , and , and and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.