Lutz Volkmann

Wiener index versus maximum degree in trees

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper, we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two different degrees.

Edge-cuts leaving components of order at least three

Let G be a graph with vertex set V(G) and edge set E(G). For X ⊆ V(G) let G[X] be the subgraph induced by X,X = V(G) - X, and (X,X) the set of edges in G with one end in X and the other in X. If G is a connected graph and S ⊆ E(G) such that G - S is disconnected, and each component of G - S consists of at least three vertices, then we speak of an order-3 edge-cut. The minimum cardinality |S| over all order-3 edge-cuts in G is called the order-3 edge-connectivity, denoted by λ3=λ3(G). A connected graph G is λ3-connected, if λ3(G) exists. An order-3 edge-cut S in G is called a λ3-cut, if |S| = λ3. First of all, we characterize the class of graphs which are not λ3-connected. Then we show for λ3-connected graphs G that λ3(G) ≤ ξ3(G), where ξ3(G) is defined by ξ3(G) = min{|(X,X)|: X ⊆ V(G),|X| = 3, G[X] is connected}. A λ3-connected graph G is called λ3-optimal, if λ3(G) = ξ3(G). If (X,X) is a λ3-cut, then X ⊆ V(G) is called a λ3-fragment. Let r3(G) = min{|X|:X is a λ3-fragment of G}. We prove that a λ3-connected graph G is λ3-optimal if and only if r3(G) = 3. Finally, we study the λ3-optimality of some graph classes. In particular, we show that the complete bipartite graph Kr,s with r,s ≥ 2 and r + s ≥ 6 is λ3-optimal.

Meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen

Uber diese Arbeit wurde auf einem “Tag der Funktionenthcorie” am 15. und 16.6.1984 an der RWTH Aachen vom zweiten Verfasser berichtet. Recently E. Mues and N. Steinmetz and independently G. G. Gundersen proved that if a nonconstant meromorphic function f shares two different non-zero values a and b (counting multiplicities) with its first derivative, then f′=f.ln this paper the following theorem is proved: If a nonconstant meromorphic function f shares a finite non-zero value (counting multiplicities) with f′ and f″ then f′ = f.In the case of an entire function the conditions can be weakened considerably.

Sufficient conditions for λ′-optimality in graphs of diameter 2

Abstract For a connected graph G the restricted edge-connectivity λ′(G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in G−S. We call a graph G λ′-optimal, if λ′(G)=ξ(G), where ξ(G) is the minimum edge-degree in G. In 1999, Wang and Li (J. Shanghai Jiaotong Univ. 33(6) (1999) 646) gave a sufficient condition for λ′-optimality in graphs of diameter 2. In this paper, we weaken this condition, and we present some related results. Different examples will show that the results are best possible and independent of each other and of the result of Wang and Li.

Cycles in multipartite tournaments: results and problems

A tournament is an orientation of a complete graph, and in general a multipartite tournament is an orientation of a complete n-partite graph. Many results about cycles in tournaments are known, but closely related problems involving cycles in multipartite tournaments have received little attention until recently. We describe some of the rapid progress in recent years on this topic, including powerful new methods and techniques. This study give rise to various interesting problems and conjectures.

Extremal Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

New bounds on the k-domination number and the k-tuple domination number

Abstract Let G = ( V , E ) be a graph, k ∈ N and let N G ( v ) and d G ( v ) denote the neighbourhood and degree of a vertex v ∈ V in G , respectively. The minimum cardinality of a set D ⊆ V with | N G ( v ) ∩ D | ≥ k for all v ∈ V ∖ D is the k -domination number γ k ( G ) of G . Similarly, the minimum cardinality of a set D ⊆ V with | ( N G ( v ) ∪ { v } ) ∩ D | ≥ k for all v ∈ V is the k -tuple domination number γ × k ( G ) of G . Let G be a graph of order n and minimum degree δ and let k ∈ N . We prove that if δ + 1 ln ( δ + 1 ) ≥ 2 k , then γ k ( G ) ≤ n δ + 1 ( k ln ( δ + 1 ) + ∑ i = 0 k − 1 1 i ! ( δ + 1 ) k − 1 − i ) and γ × k ( G ) ≤ n δ + 1 ( k ln ( δ + 1 ) + ∑ i = 0 k − 1 ( k − i ) i ! ( δ + 1 ) k − 1 − i ) . Furthermore, we prove that if δ ≥ 2 , then γ × 3 ( G ) ≤ n δ − 1 ( ln ( δ − 1 ) + ln ( ∑ v ∈ V d G ( v ) + 1 2 ) − ln ( n ) + 1 ) which generalizes a recent result of J. Harant and M. Henning.

Cycles in multipartite tournaments

An n-partite tournament is an orientation of a complete n-partite graph. We show that if D is a strongly connected n-partite (n ≥ 3) tournament, then every partite set of D has at least one vertex, which lies on an m-cycle for all m in {3, 4,..., n}. This result extends those of Bondy (J. London Math. Soc.14 (1976), 277-282) and Gutin (J. Combin. Theory Ser. B58 (1993), 319-321).

Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree

A graph is called Class 1 if the chromatic index equals the maximum degree. We prove sufficient conditions for simple graphs to be Class 1. Using these conditions we improve results on some edge-coloring theorems of Chetwynd and Hilton. We also improve a theorem concerning the 1-factorization of regular graphs of high degree.

The k-rainbow domatic number of a graph

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, . . . , fd} of k-rainbow dominating functions on G with the property that ∑ d i=1 |fi(v)| ≤ k for each v ∈ V (G), is called a k-rainbow dominating family (of functions) on G. The maximum number of functions in a krainbow dominating family on G is the k-rainbow domatic number of G, denoted by drk(G). Note that dr1(G) is the classical domatic number d(G). In this paper we initiate the study of the k-rainbow domatic number in graphs and we present some bounds for drk(G). Many of the known bounds of d(G) are immediate consequences of our results.