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The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n − lg n − 1 2 lg lg n + O(1). The proper leafage l * (G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l * (G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.

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A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.

This paper addresses the problem of an efficient dispatch mechanism in an object-oriented system with multiple inheritance. The solution suggested is a direct table indexed branch such as is used in C++. The table slot assignments are made using a coloring algorithm. The method is applicable to strongly typed languages such as C++ (with multiple inheritance added) and Eiffcl, and in a slightly slower form to less strongly typed languages like Objective C.