On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

Abstract

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to the permutation of the colors. For a plane graph G, two faces f 1 and f 2 of G are adjacent ( i , j )-faces if d ( f 1 ) = i, d ( f 2 ) = j, and f 1 and f 2 have a common edge, where d ( f ) is the degree of a face f. In this paper, we prove that every uniquely three-colorable plane graph has adjacent ( 3 , k )-faces, where k ≤ 5. The bound of five for k is the best possible. Furthermore, we prove that there exists a class of uniquely three-colorable plane graphs having neither adjacent ( 3 , i )-faces nor adjacent ( 3 , j )-faces, where i , j are fixed in { 3 , 4 , 5 } and i ≠ j. One of our constructions implies that there exists an infinite family of edge-critical uniquely three-colorable plane graphs with n vertices and 7 3 n - 14 3 edges, where n ( ≥ 11 ) is odd and n ≡ 2 ( mod 3 ).