Some Conjectures in Chromatic Topological Graph Theory Joan P. Hutchinson Macalester College, emerita Thanks to Stan Wagon for the colorful graphics. Ñ 1 2 3 4 5 6 7 ALMOST FOUR-COLORING ON SURFACES: ALMOST FOUR-COLORING ON SURFACES: #1. M.O. Albertson’s favorite CONJECTURE (1980): All vertices of a triangulation of the torus can be 4-col- ored except for at most three vertices. Motivation: Every graph on the torus can be 7-colored, and every 7-chromatic toroidal graph contains K7, a trian- gulation of the torus. Every 6-chromatic toroidal graph contains one of four graphs. [Thomassen 1994] The list for 5-chromatic toroidal graphs is infinite... An affirmative answer to this conjecture implies the Four Color Theorem. In Jensen & Toft, Graph Coloring Problems, Albertson’s Four-Color Problem. CONJECTURE: For every surface S, there is an integer f HSL such that all but f HSL vertices of a graph embed- dable on S can be 4-colored. TWO & THREE-COLORING ON SURFACES #2. The “easiest” coloring result states that every plane graph can be two-colored provided every face is bounded by an even number of edges. What is true on surfaces? Locally bipartite (aka evenly embedded) graphs are those embedded on a surface with all faces bounded by an even number of edges. There is a Heawood/Ringel type theorem for locally bipar- tite graphs on surfaces. The more “modern” question asks about locally planar, locally bipartite graphs on surfaces: The more “modern” question asks about locally planar, locally bipartite graphs on surfaces: Locally planar (Albertson & Stromquist 1980) means that all noncontractible cycles are long, as long as needed for the conjecture/question/theorem at hand. Thm. (H 1995) A locally planar, locally bipartite graph on an orientable surface can be three-colored. The chromatic number of locally planar, locally bipartite graphs on nonorientable surfaces have been determined by Mohar & Seymour, 1999. QUESTION (Kawarabayashi) Can locally planar, locally bipartite graphs be 3-list-colored? Or 4-list-colored? Theorem (Thomassen 1993) A locally planar graph on any surface can be 5-colored. Theorem (DeVos, Kawarabayashi, Mohar 2008; Postle & Thomas 2012) A locally planar graph on any surface can be 5-list-colored. PLANAR THREE-COLORING #3. The “next easiest” coloring result states that a pla- nar triangulation can be three-colored if and only if every vertex has even degree (Heawood’s theorem). Theorem (Król 1973) A planar graph is three-colorable if and only if it is a subgraph of an even-degree-d triangula- tion. ; ; QUESTION: Is the chromatic number of great-circle graphs (no three meet at a point) on the sphere three? (Felsner, Hurtado, Noy, Streinu, 2000) S. Felsner, F. Hurtado, M. Noy, and I. Streinu, Hamiltonicity and coloring properties of arrangement graphs, in Proc. Symp. on Disc. Algorithms, ACM Press H2000L 655 664. It’s “easy” to show that the upper hemisphere of great- circles can be three-colored (Kocay & Wagon) using the easy proof that the graph of straight lines in the plane (no three through a point) can be three-col- ored. But can the two hemispheres be meshed to be three- colored? PLANAR LIST-COLORING: 8c,d< 8a,b,c,d< 8a,b,c,d< 8a,b< 8a,b< 8a,b,c,d< 8a,b,c,d< 8c,d< 8c,d< 8a,b,c,d< 8a,b,c,d< 8a,b< #4. QUESTION (B. Richter): If G ¹ K4 is planar, 3-con- nected and for each v L HvL ³ min 8deg HvL, 6<, is G L- list-colorable? Thomassen asked whether there is any k such that L HvL ³ min 8deg HvL, k<, then planar, 3-con- nected G is L-list-colorable. We know G can be L-list-colored when L HvL ³ deg HvL for every v, provided G is not a Gallai tree (Borodin 1977). K5 We know G can be 5-list-colored (Thomassen 1994). Question (B. Richter): If G ¹ K4 is planar, 3-connected, and for each v L HvL ³ min 8degHvL, k<, is G L-list-col- orable when k = 6? Yes for outerplanar graphs (H 2008) Best results due to Cranston, Pruchnewski, Tuza, Voigt (J. Graph Theory 2012) They ask for which Hr, kL, G a K5-minor-free, r-con- nected graph that is not a Gallai tree is G L-list-col- orable whenever L HvL ³ min 8deg HvL, k<. PLANAR PRE-COLORING EXTENDING TO A LIST-COL- ORING: #5. Theorem (Thomassen 1994) If G is a plane graph with a distinguished face F , if two consecutive vertices of F are precolored, if all other vertices of F have 3-lists, and if all other vertices have 5-lists, then G can be list-colored. 5-lists 3-lists What if the precolored vertices are not adjacent? 5-lists 3-lists No, even for outerplanar graphs. Nor with a 1- and a 2-list: 1-list 8G,B< Yes, for 2-connected, outerplanar graphs with two 2- lists. 2-list 2-list QUESTION If G is a plane graph with a distinguished face F , if two vertices of F have 2-lists, if all other vertices of F have 3-lists, and if all other vertices have 5-lists, can G be list-colored? Yes by Postle & Thomas, 2012. And their work leads toward their solution of another question: Albertson QUESTION (1998): Is there a d > 0 such that in a planar graph G, with 5-lists on each vertex, if a set of vertices, mutually at distance at least d apart, are precolored, the precoloring extends to a list-coloring of G? YES with d = 20, 000 (Dvořák, Lidický, Mohar & Postle, ms 2011; independently Postle & Thomas 2012). Albertson answered his question in the affirmative for 5- coloring. Theorem (A&H 2002) If G is planar, G@PD is bipartite with dist HPL ³ 8, a 5-coloring of G@PD in which each compo- nent is 2-colored extends to a 5-coloring of G. #6 QUESTION (2013): Is there a d > 0 such that in a planar graph G, if G@PD is bipartite with distance between components at least d , a 2-coloring of each com- ponent of G@PD extends to a 5-list-coloring of G? (Before working on this, check with Luke Postle who has some partial results.) References Mma figs. 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