Closures and heavy pairs for hamiltonicity

Abstract

We say that a graph $G$ on $n$ vertices is $\{H,F\}$-$o$-heavy if every induced subgraph of $G$ isomorphic to $H$ or $F$ contains two nonadjacent vertices with degree sum at least $n$. Generalizing earlier sufficient forbidden subgraph conditions for hamiltonicity, in 2012, Li, Ryjáček, Wang and Zhang determined all connected graphs $R$ and $S$ of order at least 3 other than $P_3$ such that every 2-connected $\{R,S\}$-$o$-heavy graph is hamiltonian. In particular, they showed that, up to symmetry, $R$ must be a claw and $S\in \{P_4,P_5,C_3,Z_1,Z_2,B,N,W\}$. In 2008, Čada extended Ryjáček’s closure concept for claw-free graphs by introducing what we call the $c$-closure for claw-$o$-heavy graphs. We apply it here to characterize the structure of the $c$-closure of 2-connected $\{R,S\}$-$o$-heavy graphs, where $R$ and $S$ are as above. Our main results extend or generalize several earlier results on hamiltonicity involving forbidden or $o$-heavy subgraphs.