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By Cai M.-C., Favaron O., Li H.

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Extra resources for (2,k)-Factor-Critical Graphs and Toughness

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Then C-a is an Euler directed trail from x to y in G. 1 The necessary condition for an undirected graph to be eulerian was first found by Euler in 1736 when he studied the Königsberg seven bridges problern, while the sufficient condition was found in 1873 by Hierholzer (93]. We can state this as the following corollary. 8. EULERIAN GRAPHS 49 Corollary 1. 2 An undirected graph is eulerian if and only if it is connected and has no vertex of odd degree. Proof Suppose that G is an eulerian undirected graph and C is an Euler circuit of G.

Suppose v 2:: 3 below, and suppose to the contrary that P is not a Rarnilton directed path. Then n < v, and there exists some x E V(T) \ V(P) such that (x, Xn), (xi, x) E E(T). 15). It follows that (x1,x2, · · · ,Xi-I,X,xi,Xi+I, · · · ,xn) is a directed path whose length is Ionger than P's. This contradicts to the hypothesis that P is a Iongest directed path in T. Thus Pisa Rarnilton directed path in T. 2 Two vertices x and y of G are said to be connected if there is an xy-path in G. It is easy to see that "to be connected" is an equivaience relation on V(G).

6). 6. 1 Prove that dc(x, z) ~ da(x, y) + dc(y, z) for any three vertices x, y and z of a strongly connected digraph G. 2 Prove that if G is a (ß, k)-Moore digraph, then G is simple and ßregular, contains no cycle of length at most k and there is only unique (x, y)-path oflength at most k for any pair (x, y) of vertices in G. 3 Let L( G) be the line digraph of a digraph G of order v 2 2. 4 are true if G is undirected. 4 Let G be a simple undirected graph. Prove that (a) if Gis disconnected, then d(Gc) (b) if d(G) > 3, then d(Gc) < 3; (c) if d(G) = 2 and ß(G) = v- 2, ~ 2; then c 2 2v- 4.

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