The face pair of planar graphs Original Research Article

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 image is the girth pair of graph G if the shortest odd and even cycles of G have lengths image and image, respectively, and denote the number of vertices in the image-cage by image. 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 image and the bounds for the left cases. In this paper, we show the values of image for the left cases where image, image.