Planar Hamiltonian chordal graphs are cycle extendable

A cycle C in a graph G is extendible if there exists a cycle C 0 in G such that V (C) V (C 0) and jV (C 0)j = jV (C)j + 1. A graph G is cycle extendible if G contains at least one cycle and every nonhamiltonian cycle in G is extendible. A graph G is fully cycle extendible if G is cycle extendible and every vertex lies on a 3-cycle of G. G. Hendry asked if every hamiltonian chordal graph is fully cycle extendible. In this paper, we prove that every planar hamiltonian chordal graph is fully cycle extendible. All graphs we consider are simple. Deenition. A cycle C in a graph G is extendible (in G) if there exists a cycle C 0 in G such that V (C) V (C 0) and jV (C 0)j = jV (C)j + 1. If such a cycle C 0 exists we will say that C can be extended to C 0 or that C 0 is an extension of C. Deenition. A graph G is cycle extendible if G contains at least one cycle and every nonhamiltonian cycle in G is extendible. Deenition. A graph G is fully cycle extendible if G is cycle extendible and every vertex of G lies on a triangle of G. In 1], Hendry raised the following problem. Open Problem 1. Is every hamiltonian chordal graph fully cycle extendible? In this paper, we prove that every planar hamiltonian chordal graph is fully cycle extendible. Given a cycle C in a planar graph G, we specify the \forward" direction to be clockwise along the cycle. If u; v are two vertices on C , then C u; v] denotes the u; v-path on C from u to v along the speciied direction. Let C (u; v) = C u; v] ? fu; vg. If L is a path and x; y are two vertices on L, then Lx; y] denotes the subpath of L from x to y. Let L(x; y) = Lx; y] ? fx; yg. Lemma 1. Suppose G is a chordal graph and C is a cycle in G. If uv is an edge on C , then u; v have a common neighbor on C. Proof. The subgraph F of G induced by V (C) is chordal, and uv lies on at least one cycle in F. The shortest …