Algorithms and Complexity (Internet edition, 1994) - download pdf or read online

By Herbert S. Wilf

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Extra info for Algorithms and Complexity (Internet edition, 1994)

Example text

In fact, the theorem applies also to multigraphs, which are graphs except that they are allowed to have several different edges joining the same pair of vertices. 1. A (multi-)graph has an Eulerian circuit (resp. path) if and only if it is connected and has no (resp. has exactly two) vertices of odd degree. Proof: Let G be a connected multigraph in which every vertex has even degree. We will find an Eulerian circuit in G. The proof for Eulerian paths will be similar, and is omitted. The proof is by induction on the number of edges of G, and the result is clearly true if G has just one edge.

17. Devise an algorithm that will decide if a given graph, of n vertices and m edges, does or does not contain a triangle, in time O(max(n2 , mn)). 18. Prove that the number of labeled graphs of n vertices all of whose vertices have even degree is equal to the number of all labeled graphs of n − 1 vertices. , if n is a positive integer. The first definition is nonrecursive, the second is recursive. (1) ‘n! ’ (2) ‘If n = 1 then n! = 1, else n! ’ Let’s concentrate on the second definition. At a glance, it seems illegal, because we’re defining something, and in the definition the same ‘something’ appears.

Fig. 7(a) Fig. 1. Let v and w be two vertices of G such that e = (v, w) ∈ E(G). Then the number of proper K-colorings of G − {e} in which v and w have the same color is equal to the number of all proper colorings of the graph G/{e}. Proof: Suppose G/{e} has a proper K-coloring. Color the vertices of G − {e} itself in K colors as follows. Every vertex of G − {e} other than v or w keeps the same color that it has in the coloring of G/{e}. Vertex v and vertex w each receive the color that vertex vw has in the coloring of G/{e}.

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Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf

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