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9 Some linear algebra e Fig. 3. The fundamental cut De Proof . Since an edge e ∈ T lies in De but not in De for any e = e, the cut De cannot be generated by other fundamental cuts. 3). Similarly, an edge e ∈ E E(T ) lies on Ce but not on any other fundamental cycle; so the fundamental cycles form a linearly independent subset of C, of size m − n + 1. 3) m−n+1. 5 and (†), so the two inequalities above can hold only with equality. Hence the sets of fundamental cuts and cycles are maximal as linearly independent subsets of C ∗ and C, and hence are bases.

Formally, GS is the graph with vertex set S ∪ CG−S and edge set { sC | ∃ c ∈ C : sc ∈ E }; see Fig. 3. Every graph G = (V, E) contains a vertex set S with the following two properties: (i) S is matchable to CG−S ; (ii) Every component of G − S is factor-critical. Given any such set S, the graph G contains a 1-factor if and only if |S| = |CG−S |. 1 except for the—permitted—case that S or CG−S is empty factorcritical matchable GS 42 2. Matching, Covering and Packing For any given G, the assertion of Tutte’s theorem follows easily from this result.

4 obtained by replacing average with minimum degree. Deduce that |G| n0 (d/2, g) for every graph G as given in the theorem. + Show that every connected graph G contains a path of length at least min { 2δ(G), |G| − 1 }. + Find a good lower bound for the order of a connected graph in terms of its diameter and minimum degree. − Show that the components of a graph partition its vertex set. − Show that every 2-connected graph contains a cycle. 11. Determine κ(G) and λ(G) for G = P m , C n , K n , Km,n and the ddimensional cube (Exercise 2); d, m, n 3.

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Graph Theory III by Reinhard Diestel

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