By Reinhard Diestel

ISBN-10: 3540609180

ISBN-13: 9783540609186

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**Example text**

Nr ; if n1 = . . = nr =: s, we abbreviate this to Ksr . ) Graphs of the form K1,n are called stars. ,nr Ksr star = Fig. 2. Three drawings of the bipartite graph K3,3 = K32 Clearly, a bipartite graph cannot contain an odd cycle, a cycle of odd length. 1. A graph is bipartite if and only if it contains no odd cycle. 2 ] Proof . Let G = (V, E) be a graph without odd cycles; we show that G is bipartite. Clearly a graph is bipartite if all its components are bipartite or trivial, so we may assume that G is connected.

3, T has n − 1 edges, so m − n + 1 edges of G lie outside T . For each of these m − n + 1 edges e ∈ E E(T ), the graph T + e contains a cycle Ce (see Fig. 1 (iv)). Since none of the edges e lies on Ce for e = e, these m − n + 1 cycles are linearly independent. 3). Since none of the edges e ∈ T lies in De for e = e, these n − 1 cuts are linearly independent. 3) 24 1. The Basics e Fig. 3. The cut De incidence matrix The incidence matrix B = (bij )n×m of a graph G = (V, E) with V = { v1 , . . , vn } and E = { e1 , .

Show that the elements of the cycle space of a graph G are precisely the unions of the edges sets of edge-disjoint cycles in G. 28 1. The Basics 25. Given a graph G, ﬁnd among all cuts of the form E(v) a basis for the cut space of G. 26. Prove that the cycles and the cuts in a graph together generate its entire edge space, or ﬁnd a counterexample. 27. 6 generate the cycle space. 28. 6 generate the cut space. 29. What are the dimensions of the cycle and the cut space of a graph with k components?

### Graph Theory by Reinhard Diestel

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