By Shi M., Yuan X., Cai M.

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**Extra info for (3,k)-Factor-Critical Graphs and Toughness**

**Example text**

3 Modifying Graphs . . . . . . . . . . . . . . . . Vertex Identification and Edge Contraction . . . . Vertex Splitting and Edge Subdivision . . . . . . . 4 Decompositions and Coverings . . . . . . . . . . Decompositions . . . . . . . . . . . . . . . . . . . Proof Technique: Linear Independence . . . . . . . Coverings . . . . . . . . . . . . . . . . . . . . . . 5 Edge Cuts and Bonds . . . .

Another drawing of the Heawood graph b) Deduce that the Heawood graph is vertex-transitive. 2 Show that the following three graphs are isomorphic: the intersection graph of the Desargues conﬁguration, the line graph of K5 , the complement of the Petersen graph. 3 Show that the line graph of K3,3 is self-complementary. 19 is a line graph. 5 Let H := (V, F ) be a hypergraph. The number of edges incident with a vertex v of H is its degree, denoted d(v). A degree sequence of H is a vector d := (d(v) : v ∈ V ).

Det(AL(G) − xI) = (−1)m−n (x + 2)m−n det(AG − (x + 2 − k)I) c) Deduce that: i) to each eigenvalue λ = −k of G, there corresponds an eigenvalue λ + k − 2 of L(G), with the same multiplicity, ii) −2 is an eigenvalue of L(G) with multiplicity m − n + r, where r is the multiplicity of the eigenvalue −k of G. ) (H. 12 Sperner’s Lemma Let T be a triangle in the plane. A subdivision of T into triangles is simplicial if any two of the triangles which intersect have either a vertex or an edge in common.