The face pair of planar graphs

Abstract

Harary and Kovacs [Smallest graphs with given girth pair, Caribbean J. Math. 1 (1982) 24–26] have introduced a generalization of the standard cage question— r-regular graphs with given odd and even girth pair. The pair ( ω , ε ) is the girth pair of graph G if the shortest odd and even cycles of G have lengths ω and ε , respectively, and denote the number of vertices in the ( r , ω , ε ) -cage by f ( r , ω , ε ) . Campbell [On the face pair of cubic planar graph, Utilitas Math. 48 (1995) 145–153] looks only at planar graphs and considers odd and even faces rather than odd and even cycles. He has shown that f ( 3 , ω , 4 ) = 2 ω and the bounds for the left cases. In this paper, we show the values of f ( r , ω , ε ) for the left cases where ( r , ω , ε ) ∈ { ( 3 , 3 , ε ) , ( 4 , 3 , ε ) , ( 5 , 3 , ε ) , ( 3 , 5 , ε ) } .