Theorem 1 (Whitney 1927) A connected graph G with at least three vertices is 2-connected iff for every two vertices x
Show that every 2-connected graph contains at least one cycle. 9. Show that for every graph G ?(G) ? ?(G). 10. True or false? If G has no bridges
Fleischner's theorem says that the square of every 2- connected graph contains a Hamiltonian cycle. We present a proof resulting in an O(1E1) algorithm for
11) Show that every cubic 3-edge-connected graph is 3-connected. Show that every k connected graph (k ? 2) with at least 2k vertices contains a cycle.
if for every pair of vertices x and y
21-484 Graph Theory. SOLUTIONS (hbovik) - Q. 4 Diestel 3.21: Let k ? 2. Show that every k-connected graph of order at least 2k contains a cycle.
Faces and Cycles. • Theorem: – Let G be a 2-vertex-connected planar graph. Then every face in any planar drawing of G is a region of some cycle of G.
Abstract. Fleischner's theorem says that the square of every 2-connected graph contains a Hamiltonian cycle. We present a proof resulting in an O(1E1).
every 3-connected graph non-isomorphic to K4 contains a contractible edge. tractible edges in longest paths and longest cycles in 2-connected graphs.
Theorem 1 (Whitney 1927) A connected graph G with at least three vertices is 2-connected iff for every two vertices x
Theorem 1 (Whitney 1927) A connected graph G with at least three vertices is 2-connected i? for every two vertices xy ? V(G) there is a cycle containing both Proving ? (su?cient condition): If every two vertices belong to a cycle no removal of one vertex can disconnect the graph
De?nition 23 A path in a graph is a sequence of adjacent edges such that consecutive edges meet at shared vertices A path that begins and ends on the same vertex is called a cycle Note that every cycle is also a path but that most paths are not cycles Figure 34 illustrates K 5 the complete graph on 5 vertices with four di?erent
4 2 10 Theorem A graph is 2-connected iff it has a closed-ear decomposition and every cycle in a 2-edge-connected graph is the initial cycle in some such decomposition The proof of Theorem 4 2 10 is quite similar to that of Theorem 4 2 8 (with 2-connected iff ear decomposition) See p164 (=>) Show 2-edge-connectectedness is maintained on
Solutions to Homework #3: 7) Show without Menger’s theorem that every two vertices in a 2-connected graph lie on a common cycle Solution: It su?ces to show that for any two verticesx;y ofGthere are two internally vertex disjointx¡ypaths Let us show this by induction ond=dist(u;v)
De nition 1 A simple graph that has a Hamiltonian cycle is called aHamiltonian graph We observe that not every graph is Hamiltonian; for instance it is clear that a dis-connected graph cannot contain any Hamiltonian cycle/path There are also connectedgraphs that are not Hamiltonian
In Figure 2 we show a 2-connected graph G2 and paths Sand Tjoining vertices uand vof G2 such that d G2(u;v) = 2 and d P(G2 uv)(S;T) = 4 For any positive integer k>2 the graph G2 can be extended to a graph Gk such that d Gk(u;v) = kand that the diameter of P(Gk uv)) is 2k This shows that Theorem 2 is tight Figure 2: Graph G2 and paths Sand T