Degree and neighborhood conditions for hamiltonicity of claw-free graphs

<p> For a graph H , let &sigma; t ( H ) = min { &Sigma; i = 1 t d H ( v i ) | { v 1 , v 2 , &hellip; , v t } is an independent set in H } and let U t ( H ) = min { | &xcup; i = 1 t N H ( v i ) | | { v 1 , v 2 , &ctdot; , v t } is an independent set in H } . We show that for a given number &varepsilon; and given integers p &ge; t &gt; 0 , k &isin; { 2 , 3 } and N = N ( p , &varepsilon; ) , if H is a k -connected claw-free graph of order n &gt; N with &delta; ( H ) &ge; 3 and its Ryj&aacute;c̆ek&rsquo;s closure c l ( H ) = L ( G ) , and if d t ( H ) &ge; t ( n + &varepsilon; ) ∕ p where d t ( H ) &isin; { &sigma; t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p &minus; 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large:</p><p> (i) If k = 2 , &delta; ( H ) &ge; 3 , and for a given t ( 1 &le; t &le; 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s &ge; 1 ). When t = 4 and d t ( H ) = &sigma; 4 ( H ) , this proves a conjecture in Frydrych (2001).</p><p> (ii) If k = 3 , &delta; ( H ) &ge; 24 , and for a given t ( 1 &le; t &le; 10 ) d t ( H ) &gt; t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved &sigma; t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p &minus; 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer.</p>