Hansjoachim Walther

Shortness exponents of families of graphs

Abstract Known estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions.

Chromatic number of prime distance graphs

For any set D of positive integers, the distance graph G(D)G(V,E) is the graph with vertex set V(G)Z and edge set E(G){(u,v):|u−v|∈D}. In Research Problem 77 (Discrete Math. 69 (1988) 105–106) Eggleton, Erdos and Skilton propose the problem to determine all minimal subsets D of the prime numbers such that graph G(D) is 4-chromatic. In the present paper this problem is solved for 4-element prime sets D.

Über die Nichtexistenz eines Knotenpunktes, durch den alle längsten Wege eines Graphen gehen

Abstract Es wird ein zusammenhangender Graph konstruiert, der keinen Knotenpunkt besitzt, durch den alle langsten Wege des Graphen gehen. In einer spateren Arbeit wird ein 3-fach zusammenhangender Graph G (bzw. einfach zusammenhangender Graph H) angegeben, der keine zwei Knotenpunkte besitzt, so das in jedem langsten Kreis von G (bzw. in jedem langsten Weg von H) mindestens einer der beiden Knotenpunkte liegt. Damit sind zwei von T. Gallai und H. Sachs auf dem Kolloquium uber Graphentheororie 1966 in Tihany (Ungarn) [2] gestellte Probleme gelost.

On vertex-degree restricted paths in polyhedral graphs

Abstract It is proved that every 3-connected planar graph G with δ ( G )⩾4 either does not contain any path on k ⩾8 vertices or must contain a path on k vertices ( k ⩾8) having degree (in G) at most 5 k −7; the bound 5 k −7 is shown to be the best possible. For every connected planar graph H different from a path and for every integer m ⩾4 there is a 3-connected planar graph G with δ ( G )⩾4 such that each subgraph of G isomorphic to H has a vertex x with deg G ( x )⩾ m .

On the chromatic number of special distance graphs

Abstract For all l ⩾ 10 and u ⩾ l 2 − 6l + 3 the chromatic number is proved to be 3 for distance graphs with all integers as vertices, and edges only if the vertices are at distances 2, 3, u, and u + l.

On the radius of graphs

Let G be any 3-connected graph containing n vertices and r the radius of G. Then the inequality r < 14n + O(log n) is proved. A similar theorem concerning any (2m −1)-connected graph G can be proved too.

Polyhedral graphs with restricted number of faces of the same type

Let G = G(V,E,F) be a polyhedral graph with vertex set V, edge set E and face set F. A face α is an 〈a1,.....,al〉-face if α is an l-gon and the degrees d(xi) of the vertices x1,.....,xl incident with α in the cyclic order are a1,....,al, respectively. The lexicographic minimum 〈b1,....,bl〉 such that α is a 〈b1,...,bl〉-face is called the type of α. Furthermore let z be a given integer. We consider polyhedral graphs where the number of faces of each type is restricted by z. We prove that there is only a finite number of such graphs.

Über die Nichtexistenz zweier Knotenpunkte eines Graphen, die alle längsten Kreise fassen

Zusammenfassung Es wird ein zweifach zusammenhangender (nichtplanarer) Graph angegeben, der keine zwei Knotenpunkte besitzt, so das jeder langste Kreis des Graphen durch weinigstens einen der beiden Knotenpunkte geht.

Polyhedral graphs with extreme numbers of types of faces

A face α ∈ F of a polyhedral graph G(V,E,F) is an (a1,a2 ..... al)-face if α is an l-gon and the degrees d(xi) of the vertices xi ∈ V incident with α in the cyclic order are ai, i = 1,2,...,l. The lexicographic minimum 〈b1,b2 ..... bl〉 such that α. is a (b1,b2 ..... bl)-face is the type of α. All polyhedral graphs having only one type of faces are listed. It is proved that the set of triangulations having only faces of different types is non-empty and finite.

Vertex-oblique graphs

Let x be a vertex of a simple graph G. The vertex-type of x is the lexicographically ordered degree sequence of its neighbors. We call the graph G vertex-oblique if there are no two vertices in V(G) which are of the same vertex-type. We will show that the set of vertex-oblique graphs of arbitrary connectivity is infinite.