Graph_Theory_Notes7.pdf

# Denote by v and v the predecessor and successor of v

• Notes
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Denote by v - and v + the predecessor and successor of v , respectively. Note that v - is undefined if v is the first vertex of P , and v + is undefined if v is the last vertex of P . For any subset S of V ( P ) or V ( C ), set S - = { v - : v S } , S + = { v + : v S } . Path exchanges Let P be an ( x, y )-path in a graph G . If z is a vertex adjacent to y but not on P , then we can extend P to get a longer path by including yz . If z is a vertex on P adjacent to y , but z 6 = y - , then we can transform P to a new path P 0 starting from x by adding the edge yz and deleting the edge zz + . This transformation from P to P 0 is called a path exchange . Of course, P 0 has the same length as P . However, if it happens that z + is adjacent to a vertex not on P 0 , then we P 0 can be extended to a longer path. Cycle exchanges Let C be a cycle in a graph G . If there are nonconsecutive vertices x and y on C such that both xy and x + y + are edges of G , then by adding these two edges and deleting xx + and yy + , we obtain a new cycle C 0 of the same length. The transformation from C to C 0 is called a cycle exchange . 3 Sufficient conditions Dirac’s Theorem Theorem 2 (Dirac, 1952) . Let G be a graph with order n 3 and minimum degree δ ( G ) n/ 2. Then G is Hamiltonian. Proof. Consider the complete graph K = G G , with the edges of G coloured blue and the edges of G coloured red. Since K is a complete graph, it is Hamiltonian. Let C be a Hamilton cycle of K with as many blue edges as possible. We claim that all edges of C are blue, that is, C is a Hamilton cycle of G . Proof (continued). Suppose otherwise. Let xx + be a red edge of C , where x + is the successor of x on C . Set S = N G ( x ) , T = N G ( x + ) . Then | S + | + | T | = | S | + | T | = deg( x ) + deg( x + ) 2 δ ( G ) n. Since x + 6∈ S + and x + 6∈ T , we have S + T V ( G ) - { x + } . Hence | S + T | = | S + | + | T | - | S + T | ≥ n - ( n - 1) = 1 . So we can take y + S + T . Let C 0 be the Hamilton cycle of K obtained by exchanging xx + and yy + for the blue edges xy and x + y + . Then C 0 has more blue edges than C , contradicting the choice of C .

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Ore’s Theorem Theorem 3 (Ore, 1960) . Let G be a graph with order n 3 such that deg( u ) + deg( v ) n for any pair of nonadjacent vertices u and v . Then G is Hamiltonian.
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• One '14
• Graph Theory, Glossary of graph theory, Graph theory objects, Hamiltonian path, Gabriel Andrew Dirac

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