Dirac’s Type Sufficient Conditions for Hamiltonicity and Pancyclicity

Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,

On Characterizations of Rigid Graphs in the Plane Using Spanning Trees

We study characterizations of generic rigid graphs and generic circuits in the plane using only few decompositions into spanning trees. Generic rigid graphs in the plane can be characterized by spanning tree decompositions [5,6]. A graph G with n vertices and 2n − 3 edges is generic rigid in the plane if and only if doubling any edge results in a graph which is the union of two spanning trees. This requires 2n − 3 decompositions into spanning trees. We show that n − 2 decompositions suffice: only edges of G − T can be doubled where T is a spanning tree of G. A recent result on tensegrity frameworks by Recski [7] implies a characterization of generic circuits in the plane. A graph G with n vertices and 2n − 2 edges is a generic circuit in the plane if and only if replacing any edge of G by any (possibly new) edge results in a graph which is the union of two spanning trees. This requires decompositions into spanning trees. We show that 2n − 2 decompositions suffice. Let be any circular order of edges of G (i.e. ). The graph G is a generic circuit in the plane if and only if is the union of two spanning trees for any . Furthermore, we show that only n decompositions into spanning trees suffice.     

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.      

On the Circumference of 2-Connected $$\mathcal{P}_{3}$$-Dominated Graphs

Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs.     

Packing Trees with Constraints on the Leaf Degree

If m is a positive integer then we call a tree on at least 2 vertices an m-tree if no vertex is adjacent to more than m leaves. Kaneko proved that a connected, undirected graph G = (V, E) has a spanning m-tree if and only if for every the number of isolated vertices of G − X is at most —unless we have the exceptional case of and m = 1. As an attempt to integrate this result into the theory of graph packings, in this paper we consider the problem of packing a graph with m-trees. We use an approach different from that of Kaneko, and we deduce Gallai–Edmonds and Berge–Tutte type theorems and a matroidal result for the m-tree packing problem. Jácint Szabó: Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No.504438.     

List Point Arboricity of Dense Graphs

Let G be a simple graph. The point arboricity ρ(G) of G is defined as the minimum number of subsets in a partition of the point set of G so that each subset induces an acyclic subgraph. The list point arboricity ρ _l (G) is the minimum k so that there is an acyclic L-coloring for any list assignment L of G which |L(v)| ≥ k. So ρ(G) ≤ ρ _l (G) for any graph G. Xue and Wu proved that the list point arboricity of bipartite graphs can be arbitrarily large. As an analogue to the well-known theorem of Ohba for list chromatic number, we obtain ρ _l (G + K _n ) = ρ(G + K _n ) for any fixed graph G when n is sufficiently large. As a consequence, if ρ(G) is close enough to half of the number of vertices in G, then ρ _l (G) = ρ(G). Particularly, we determine that , where K _2(n) is the complete n-partite graph with each partite set containing exactly two vertices. We also conjecture that for a graph G with n vertices, if then ρ _l (G) = ρ(G). Research supported by NSFC (No.10601044) and XJEDU2006S05.     

Generalized Thrackles and Geometric Graphs in $${\mathbb{R}}^3$$ with No Pair of Strongly Avoiding Edges

We define the notion of a geometric graph in . This is a graph drawn in with its vertices drawn as points and its edges as straight line segments connecting corresponding points. We call two disjoint edges of G strongly avoiding if there exists an orthogonal projection of to a two dimensional plane H such that the projections of the two edges on H are contained in two different rays, respectively, with a common apex that create a non-acute angle. We show that a geometric graph on n vertices in with no pair of strongly avoiding edges has at most 2n − 2 edges. As a consequence we get a new proof to Vázsonyi’s conjecture about the maximum number of diameters in a set of n points in . This research was supported by THE ISRELI SCIENCE FOUNDATION (grant No. 938/06).     

The Unbalance of Set Systems

The unbalance of an intersecting family is defined as , where is the maximum degree of i.e. the maximum of over all vertices x. We show that the unbalance of a k-uniform intersecting family is at most when n ≥ 6k ³ and we determine all families achieving this bound.     

Disproof of a Conjecture on the Subdivision Domination Number of a Graph

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sd_γ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Haynes et al. (Discussiones Mathematicae Graph Theory 21 (2001) 239-253) conjectured that for any graph G with . In this note we first give a counterexample to this conjecture in general and then we prove it for a particular class of graphs.     

Measures on Graphs and Groupoid Measures

The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G ₁ and G ₂ are graph-isomorphic, then the von Neumann algebras L ^∞(μ ₁) and L ^∞(μ ₂) are *-isomorphic, where μ ₁ and μ ₂ are the same kind of our graph measures of G ₁ and G ₂, respectively. Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007.     

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G be a finite abelian group with k _i |k _i-1(2 ≤ i ≤ n), and suppose that (r _i , k ₁) = 1(1 ≤ i ≤ s). Let denote the maximal cardinality of a set which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with . We prove that . We also apply this result to study problems in finite projective spaces.      

Tighter Bounds on the Size of a Maximum <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-Matching in a Cubic Graph

We establish that for any connected cubic graph G of order n > 16 the maximum P ₃-matching in G consists of at least vertices.     

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph K _n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product of graphs. As the main result of this paper, we prove that for any two graphs G ₁ and G ₂ with η(G ₁) = h and η(G ₂) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following:

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph

In this paper we study tight lower bounds on the size of a maximum matching in a regular graph. For k ≥3, let G be a connected k-regular graph of order n and let α′(G) be the size of a maximum matching in G. We show that if k is even, then , while if k is odd, then . We show that both bounds are tight. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Limit sets and strengths of convergence for sequences in the duals of thread-like Lie groups

We consider a properly converging sequence of non-characters in the dual space of a thread-like group and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if (π _k ) is a properly convergent sequence of non-characters in , then there is a trade-off between the number of limits σ which are not characters, their degrees, and the strength of convergence i _σ to each of these limits (Theorem 3.2). This enables us to describe various possibilities for maximal limit sets consisting entirely of non-characters (Theorem 4.6). In Sect. 5, we show that if (π _k ) is a properly converging sequence of non-characters in and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to ) with the limit set has at most two components. In the case of two components, each is a half-plane. In Theorem 7.7, we show that if a sequence has a character as a cluster point then, by passing to a properly convergent subsequence and then a further subsequence, it is possible to find a real null sequence (c _k ) (with ) such that, for a in the Pedersen ideal of C ^*(G _N ), exists (not identically zero) and is given by a sum of integrals over .     

A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph

Let G be a graph and S ⊂ V(G). We denote by α(S) the maximum number of pairwise nonadjacent vertices in S. For x, y ∈ V(G), the local connectivity κ(x, y) is defined to be the maximum number of internally-disjoint paths connecting x and y in G. We define . In this paper, we show that if κ(S) ≥ 3 and for every independent set {x ₁, x ₂, x ₃, x ₄} ⊂ S, then G contains a cycle passing through S. This degree condition is sharp and this gives a new degree sum condition for a 3-connected graph to be hamiltonian.     

Using Lagrangians of Hypergraphs to Find Non-Jumping Numbers (I)

Let r ≥ 2 be an integer. A real number is a jump for r if for any and any integer m, m ≥ r, any r-uniform graph with vertices and density at least contains a subgraph with m vertices and density at least α + c, where c = c(α) does not depend on or m. It follows from a result of Erdős, Stone, and Simonovits that every is a jump for r = 2. Erdőos asked whether the same is true for r ≥ 3. Frankl and R?dl gave a negative answer by showing an infinite sequence of non-jumping numbers for r ≥ 3. However, there are a lot of unknowns on determining whether a number is a jump for r ≥ 3. In this paper, we first find two infinite sequences of non-jumping numbers for r = 4, then we extend one of the results to every r ≥ 4. Our approach is still based on the approach developed by Frankl and R?dl. Received November 30, 2005     

Hamiltonian cycles and 2-dominating induced cycles in claw-free graphs

Let G = (V, E) be a connected graph. For a vertex subset , G[S] is the subgraph of G induced by S. A cycle C (a path, respectively) is said to be an induced cycle (path, respectively) if G[V(C)] = C (G[V(P)] = P, respectively). The distance between a vertex x and a subgraph H of G is denoted by , where d(x, y) is the distance between x and y. A subgraph H of G is called 2-dominating if d(x, H) ≤ 2 for all . An induced path P of G is said to be maximal if there is no induced path P′ satisfying and . In this paper, we assume that G is a connected claw-free graph satisfying the following condition: for every maximal induced path P of length p ≥ 2 with end vertices u, v it holds: