David W. Bange

Generalized domination and efficient domination in graphs

Abstract This paper generalizes dominating and efficient dominating sets of a graph. Let G be a graph with vertex set V (G). If ƒ: V (G) → Y , where Y is a subset of the reals, the weight of ƒ is the sum of ƒ(v) over all v ϵ V(G). If the closed neighborhood sum of ƒ(v) at every vertex is at least 1, then ƒ is called a Y-dominating function of G. If the closed neighborhood sum is exactly 1 at every vertex, then ƒ is called an efficient dominating function. Two Y-dominating functions are equivalent if they have the same closed neighborhood sum at every vertex of G. It is shown that if the closed neighborhood matrix of G is invertiable then G has an efficient Y-dominating function for some Y. It is also shown that G has an efficient Y-dominating function if and only if all equivalent Y-dominating functions have the same weight. Related theoretical and computational questions are considered in the special cases where Y = {−1, 1} or Y = {−1, 0, 1}.

Periodic solutions of a quasilinear parabolic differential equation

Abstract This paper treats the quasilinear, parabolic boundary value problem u xx − u t = −ƒ(x, t, u) u (0, t ) = ϑ 1 ( t ); u ( l , t ) = ϑ 2 ( t ) on an infinite strip {(x, t) ¦ 0 with the functions ƒ(x, t, u), ϑ 1 (t), ϑ 2 (t) being periodic in t . The major theorem of the paper gives sufficient conditions on ƒ(x, t, u) for this problem to have a periodic solution u ( x , t ) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on ƒ(x, t, u) and indicate a method for determining the initial estimate at which the iteration may begin.

Sequentially additive graphs

The concept of a simply sequentially additive graph is introduced as follows. A graph G with |V(G)|+|E(G)|=M is said to be simply sequentially additive if there is a bijection f:V(G)@?E(G)->{;1,2,...,M}; such that for each x = (u, @n) in E(G), f(x) = f(u) + f(@n). Various aspects of finding such numberinga or showing that none exist are discussed.

Efficient domination of the orientations of a graph

Abstract For an orientation G of a simple graph G, N G [x] denotes the vertex x together with all those vertices in G for which there are arcs directed toward x. A set S of vertices of G is an efficient dominating set (EDS) of G provided that IFld | N G [x]∩ S| = 1 for every x in G . An efficiency of G is an ordered pair ( G , S), where S is an EDS of the orientation G of G. The number of distinct efficiencies of G is denoted is denoted by η(G). We give a formula for η(G) which allows us to calculate it for complete graphs, complete bipartite graphs, cycles, and paths. We find the minimum and maximum value of η(G) among all graphs with a fixed number of edges. We also find the minimum and maximum value of η(G), as well as the external graphs, among all graphs with a fixed number of vertices. Finally, we show that the probability a random oriented graph has an EDS is exponentially small when such graph is chosen according to a uniform distribution.

An existence theorem for periodic solutions of a nonlinear parabolic boundary value problem

and assume that there are functions a(x) and P(X) satisfying oL”(x) +f(x, 4x), 44) > 0; B”(x) +.k B(x), B’(x)) Q 0 on [0, 21. Then, by use of the Leray-Schauder fixed-point theorem, existence of a solution u(x, t) is obtained under standard continuity and growth conditions on f (x, t, u, p), VI(t), and v%(t). Additional restrictions on f(x, t, u, p) are given which permit the determination of suitable CX(X) and p(x).

Class-reconstruction of total graphs

It is shown that given any vertex-deleted total graph, every reconstruction into a total graph by the addition of a vertex yields the original total graph. The proof indicates how the reconstruction can be done.

Does a Parabola Have an Asymptote

David Bange is a professor of mathematics at the University of Wisconsin-LaCrosse, where he has taught undergraduates since earning his Ph.D. at Colorado State University in 1971 under the direction of Robert E. Gaines. He also served as department chair for several years while he slowly came to his senses and a realization that the psychic rewards of teaching easily outweigh any financial rewards of administration. His primary research interests are domination problems in graph theory.

A note on similar edges and edge-unique line graphs

If G is a connected graph having no vertices of degree 2 and L(G) is its line graph, two results are proven: if there exist distinct edges e and f with L(G) − e ≅ L(G) − f then there is an automorphism of L(G) mapping e to f; if G − u ¦ G − v for any distinct vertices u, v, then L(G) − e ¦ L(G) − f for any distinct edges e, f.