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A note on intersection dimensions of graph classes

Petr Hliněný, Aleš Kuběna (1995)

Commentationes Mathematicae Universitatis Carolinae

The intersection dimension of a graph G with respect to a class 𝒜 of graphs is the minimum k such that G is the intersection of some k graphs on the vertex set V ( G ) belonging to 𝒜 . In this paper we follow [ Kratochv’ıl J., Tuza Z.: Intersection dimensions of graph classes, Graphs and Combinatorics 10 (1994), 159–168 ] and show that for some pairs of graph classes 𝒜 , the intersection dimension of graphs from with respect to 𝒜 is unbounded.

A partition of the Catalan numbers and enumeration of genealogical trees

Rainer Schimming (1996)

Discussiones Mathematicae Graph Theory

A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers C n , k of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the C n , k form a partition of the n-th Catalan numer Cₙ, that means C n , 1 + C n , 2 + . . . + C n , n = C .

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