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Note on a paper of McMorris and Shier

Heinz-Jürgen Voss — 1985

Commentationes Mathematicae Universitatis Carolinae

Graphs with prescribed maximal subgraphs and critical chromatic graphs

Heinz-Jürgen Voss — 1977

Commentationes Mathematicae Universitatis Carolinae

Sachs triangulations, generated by dessins d'enfant, and regular maps

Heinz-Jürgen Voss — 1997

Mathematica Slovaca

Trestles in polyhedral graphs

Michal Tkáč; Heinz-Jürgen Voss — 2002

Discussiones Mathematicae Graph Theory

Light classes of generalized stars in polyhedral maps on surfaces

Stanislav Jendrol'; Heinz-Jürgen Voss — 2004

Discussiones Mathematicae Graph Theory

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. $S_{i}$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(), on d + 2- m() vertices...

Light paths with an odd number of vertices in polyhedral maps

Stanislav Jendroľ; Heinz-Jürgen Voss — 2000

Czechoslovak Mathematical Journal

Let $P_{k}$ be a path on $k$ vertices. In an earlier paper we have proved that each polyhedral map $G$ on any compact $2$ -manifold $𝕄$ with Euler characteristic $χ (𝕄) \leq 0$ contains a path $P_{k}$ such that each vertex of this path has, in $G$ , degree $\leq k ⌊\frac{5 + \sqrt{49 - 24 χ (𝕄)}}{2}⌋$ . Moreover, this bound is attained for $k = 1$ or $k \geq 2$ , $k$ even. In this paper we prove that for each odd $k \geq \frac{4}{3} ⌊\frac{5 + \sqrt{49 - 24 χ (𝕄)}}{2}⌋ + 1$ , this bound is the best possible on infinitely many compact $2$ -manifolds, but on infinitely many other compact $2$ -manifolds the upper bound can be lowered to $⌊(k - \frac{1}{3}) \frac{5 + \sqrt{49 - 24 χ (𝕄)}}{2}⌋$ .

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