Triangle-free planar graphs with minimum degree 3 have radius at least 3

We prove that every triangle-free planar graph with minimum degree 3 has radius at least 3; equivalently, no vertex neighborhood is a dominating set. In 1975, Plesnik [3] determined all triangle-free planar graphs with diameter 2. They are the stars, the complete bipartite graphs K2,n, and a third family that can be described in several ways. One can start with the disjoint union K2 +K1 and add vertices of degree 2 joined to either nonadjacent pair of the original triple, or start with C5 and expand two nonadjacent vertices into larger independent sets, or start with K2,n and apply a “vertex split” to one of the high-degree vertices. Each graph in Plesnik’s characterization has a vertex of degree at most 2. Thus his result implies that every triangle-free planar graph with minimum degree 3 has diameter at least 3 (note that no triangle-free planar graph has minimum degree greater than 3). In this note, we strengthen this statement by proving that every triangle-free graph with minimum degree 3 has radius at least 3. That is, it has no vertex whose neighborhood is a dominating set. There are many triangle-free planar graphs with minimun degree 3 and radius equal to 3. Our result can also be related to other past work about distances in triangle-free or planar graphs. Erdős, Pach, Pollack, and Tuza [1] studied the maximum radius and diameter among graphs with fixed minimum degree. They also solved these problems in the family of triangle-free graphs. In contrast, we are seeking the minimum radius when the family is further restricted to planar graphs. For planar graphs, Harant [2] proved an upper bound on the radius when the graph is 3-connected and has no long faces (it is n/6 + q + 3 2 when the graph has n vertices and no face of length more than q). We prove a lower bound on the radius when the graph has no short faces (no triangles), without restriction on connectivity. Mathematics Education Department, Konkuk University, Seoul, Korea, skim12@konkuk.ac.kr Department of Mathematics, University of Illinois, Urbana, IL 61801, west@math.uiuc.edu. This research is partially supported by the National Security Agency under Award No. H98230-06-1-0065.