Hamiltonian Cycles Hamiltonian Cycles CSE, IIT KGP Hamiltonian - PowerPoint PPT Presentation

Hamiltonian Cycles Hamiltonian Cycles CSE, IIT KGP

Hamiltonian Cycle Hamiltonian Cycle • A A Hamiltonian cycle Hamiltonian cycle is a spanning cycle in a is a spanning cycle in a • graph. graph. – The c The c ircumference ircumference of a graph is the length of its of a graph is the length of its – longest cycle. longest cycle. – A A Hamiltonian path Hamiltonian path is a spanning path. is a spanning path. – – A graph with a spanning cycle is a A graph with a spanning cycle is a Hamiltonian Hamiltonian – graph . . graph CSE, IIT KGP

Necessary and Sufficient Conditions Necessary and Sufficient Conditions • [Necessary:] If G has a Hamiltonian cycle, then for If G has a Hamiltonian cycle, then for • [Necessary:] ⊆ V, the graph G − S has at most |S| any set S ⊆ V, the graph G − any set S S has at most |S| components. components. • [Sufficient: Dirac Dirac:1952 :1952 ] ] If G is a simple graph with at If G is a simple graph with at • [Sufficient: δ (G) ≥ n(G)/2, then G is least three vertices and δ (G) ≥ n(G)/2, then G is least three vertices and Hamiltonian. Hamiltonian. • [Necessary and sufficient:] If G is a simple graph and If G is a simple graph and • [Necessary and sufficient:] u,v are distinct non are distinct non- -adjacent vertices of G with adjacent vertices of G with u,v ≥ n(G), then G is Hamiltonian if and only if ) ≥ d( u u ) + d( ) + d( v v ) n(G), then G is Hamiltonian if and only if d( G + uv uv is Hamiltonian. is Hamiltonian. G + CSE, IIT KGP

Hamiltonian Closure Hamiltonian Closure The Hamiltonian closure of a graph G, denote The Hamiltonian closure of a graph G, denote C(G), is the supergraph supergraph of G on V(G) obtained by of G on V(G) obtained by C(G), is the iteratively adding edges between pairs of non- - iteratively adding edges between pairs of non adjacent vertices whose degree sum is at least n n , , adjacent vertices whose degree sum is at least until no such pair remains. until no such pair remains. – The closure of G is well The closure of G is well- -defined defined – – A simple A simple n n - -vertex graph is Hamiltonian if and vertex graph is Hamiltonian if and – only if its closure is Hamiltonian only if its closure is Hamiltonian CSE, IIT KGP

And more… And more… χ (G) ≥ α α (G), then G has a Hamiltonian If χ (G) ≥ • If (G), then G has a Hamiltonian • cycle (unless G = K 2 ) cycle (unless G = K 2 ) CSE, IIT KGP