Hadwiger number of a graph: Question about the original article from 1943

Question

I am analyzing Hadwiger's original article (Hadwiger, Hugo (1943), "Uber eine Klassifikation der Streckenkomplexe", Vierteljschr. Naturforsch. Ges. Zurich, 88: 133–143) for my work related to the Hadwiger Conjecture in graph theory.

This article is in German, and Hadwiger's terminology from 1943 is very different from current graph theory terminology. For example, he says "Komplex" for what we now call a graph, and he says "Simplex S(n)" for what we now call the complete graph $K_n$ over $n$ vertices. Also, when he says "K(k)", now we would call this a graph with Hadwiger number $k$ (for him, K(k) is not the complete graph).

In this article, he defines what we now call the Hadwiger number ${\rm had}(A)=k$, the size of the largest complete graph $K_k$ that can be obtained by contracting edges of $A$ (where $A$ is an undirected graph).

His original definition is

Ein Komplex A heisst ein K(k), wenn er sich auf einen S(k), aber nicht auf einen S(k+1) zusammenziehen lässt. Ein K(k) ist notwendig ein zusammenhängender Komplex.

I am translating this into current terminology "An undirected graph $A$ has Hadwiger number $k$, if the complete graph $K_k$, but not $K_{k+1}$, can be obtained by contracting edges of $A$. It is necessarily a connected graph."

In the next sentences he says

Die Eigenschaft, ein K(k) zu sein, kennzeichnet eine Art des höheren Zusammenhangs, die durch die natürliche Zahl k gegeben ist.

What does "höheren Zusammenhangs" mean here? My current terminology translation would be "The property to have Hadwiger number $k$ characterizes a higher degree of connectedness, which is given by the natural number $k$".

But my second, more deliberate translation sounds more logical to me: "The property to have Hadwiger number $k$ characterizes a more abstract idea of connectedness, which can just be expressed by the natural number $k$". But it is more of an interpretation than the first version.

I have asked two German native speaker scientists, but they were not sure as they are no experts in graph theory. It would be great if someone could help me with this.

Answer 1

Let me give it a try. As a disclaimer, English is not my mother tongue, so my translation might have linguistic flaws.

First of all, I would say the sentence is hard to translate and it is a bit informal, i.e. it is not a rigorous mathematical statement. In my view, this sentence gives an informal motivation why it it is useful to look a "K(k)" graphs, i.e. graphs with Hadwiger number $k$. Secondly, I think your 2nd version is relatively close to what this sentence means, but I think the meaning goes beyond connectedness.

My suggestion is to translate it in the context of the beginning of the whole paragraph where you took the quotes from. The first sentence in this paragraph reads:

"Im folgenden sprechen wir von einer Möglichkeit der Klassifikation der Streckenkomplexe, die besonders im Hinblick auf das Problem der chromatischen Zahl von besonderem Interesse zu sein scheint."
(My translation: In the following, we speak of a possible classification of graphs, which seems to be of particular interest regarding the problem of chromatic number.)

So this is setting the scene really broadly, and announcing that the following definition could be relevant for the chromatic number problem, and in particular for the famous 4-color-problem. Next, he defines the Hadwiger number (his "K(k)" graphs), and then there is the sentence you are asking about. Here is my suggested translation:

The property of having Hadwiger number $k$ ($k$ a natural number) characterizes a deeper connection between those graphs. (in the sense of: deeper than the ordinary notion of connectendness)