Some Open Problems in Graph T

Transcription

1 Some Open Problems in Graph T February 11,

2 Sources Graph Theory Open Problems - Six problems suitable f uate research projects hochberg/undopen/graphth Douglas West has compiled a list of open problems west/openp/ The archives for the book Graph Coloring Problems R. Jensen and Bjarne Toft (Wiley Interscience 1995), dedi Erdös. Unsolved Problems - Including the list of 50 problems Murty with current status. Compiled by Stephen C. Lock 0-1

3 Sources Perfect Problems chvatal/perfect/problems.h 0-2

4 Unit Distance Graphs chromatic number Open since 1956 DESCRIPTION: How many colors are needed so that if the plane is assigned one of the colors, no two points whic distance 1 apart will be assigned the same color? It is kn answer is either 4, 5, 6 or 7 this is not too hard to show try it now in order to get a flavor for what this problem is This number is also called the chromatic number of the p A graph which can be embedded in the plane so that vertic to points in the plane and edges correspond to unit-length is called a unit-distance graph. The question above is asking what the chromatic number of unit-distance graphs Here are some warm-up questions, whose answers are k complete bipartite graphs are unit-distance graphs? What 4-chromatic unit-distance graph? Show that the Peterse unit-distance graph. 0-3

5 Barnette s Conjecture DESCRIPTION: Is it true that every 3-connected, 3-regul partite graph is Hamiltonian? It is known that this is not true if you remove the biparti but the smallest known such graph which is not Hamilt vertices. 0-4

6 Crossing Number of K 9,9 DESCRIPTION: What is the crossing number of the comp graph K 9,9? It is conjectured to be 256, but nobody know Another way to ask this is the following: If you place 9 red plane and 9 blue points in the plane, and then connect eac each blue point with curves (81 curves in all), then what is number of crossing points that must appear in your drawi An example for K 4,4 is shown to the right, with 8 crossin this graph can be drawn with just 4 crossings...can you fin For these drawings, it is assumed that the curves do not tou (vertices) except at their endpoints, and that no three cur point. 0-5

7 Traversal by Prime Sum Posted 7/11/03 Named for the classical Gray code listing binary vectors (cy one bit change between successive vectors, A combinator is a listing of the objects in a set using only specified cha successive objects. The last item should also be close to the is sought is a Hamiltonian cycle in the graph defined by t adjacencies. Originator(s):???? Question: Let G m be the graph with vertex set {1, 2,..., 2 xy is an edge if and only if x + y is prime. Is G m Ham m >= 2? Comments/Partial results: It is easy to build a Hamiltonia 2m + 1 and 2m + 3 are both prime, but it is not even kn Hamiltonian for infinitely many m. 0-6

8 Total Coloring Conjecture Behzad and Vizing, 1965 Graph Coloring Problems by Tommy R. Jensen and Bja Problem 4.9: Total Coloring This is a mixture type of the ordinary vertex and edge sense that a total coloring of a graph G assigns a color to vertices and edges. A proper total coloring is one in whic are properly colored in the usual sense, likewise, the edge a vertex and edge are incident, they get different colors. Conj: Every graph G has a proper total coloring using at m colors Total coloring conjecture in the case of multigraphs with been published in: A.V. Kostochka, The total chromatic number of any mu maximum degree five is at most seven, Discrete Mathemat

9 List Total Coloring The List Total Coloring Conjecture states that the total c any graph equals its total chromatic number. 0-8

10 From Bondy and Murty An Extremal Problem P. Erdös and N. Sauer, Let f(n) be the maximum possible number of edges in a on n vertices which contains no 3-regular subgraph. Deter Since there is a constant c such that every simple graph cn8/5 edges contains the 3-cube (P. Erdös and M. Simo clearly f(n) < cn 8/5. The best known upper and lower bounds are given in the fol L.Pyber, V.Rödl, E.Szemeredi, Dense graphs without 3 graphs, JCTB 63(1995), c 1 n log log n f(n) c 2 n log n 0-9

11 Strong Edge-Coloring Originator(s): P. Erdös and J. Nešetřil (1985). R. Faudree R. Schelp, and Zs. Tuza (1989). Definition: A strong edge-coloring of a graph G is a ed which every color class is an induced matching; that is, an belonging to distinct edges with the same color are not a strong chromatic index s (G) is the minimum number of col edge-coloring of G. Conjectures: If G is a simple graph with maximum degree 1) s (G) <= 5D 2 /4 if D is even, and s (G) <= (5D2 2D is odd. 2) s (G) <= D 2 if G is bipartite. (Stronger versions descri 3) s (G) <= 7 if G is bipartite with maximum degree 4-cycle. 4) s (G) <= 9 if G is planar and 3-regular. 0-10

12 Background, motivation, and references are given: west/openp/strongedge.html 0-11

13 The Strong Perfect Graph Conjecture Claude Berge, 1960 Perfect graphs and Berge graphs: The chromatic numbe F is defined as the smallest number of colors that can b vertices of F in such a way that every two adjacent vertic distinct colors; the clique number of F is defined as the la of pairwise adjacent vertices in F = largest complete subg Trivially, the chromatic number of every graph is at least it ber. A graph G is called perfect if, for each of its induc F, the chromatic number of F equals the largest numbe adjacent vertices in F. 0-12

14 The Strong Perfect Graph Conjecture Claude Berge, 1960 A chordless cycle is an induced cycle. A hole is a chor length at least four; an antihole is the complement of such and antiholes are odd or even according to the parity of th vertices. No odd hole is perfect (the clique number of an and its chromatic number is 3) and no odd antihole is perfe number of an antihole with 2k+1 vertices is k and its chrom is k+1). Conj: A graph is perfect if (and only if) it contains no no odd antihole. This conjecture became known as the S Graph Conjecture. 0-13

15 The Strong Perfect Graph Conjecture has become the Strong Perfect Graph Theorem The Strong Perfect Graph Theorem In May 2002, Maria Chudnovsky and Paul Seymour announ building on earlier joint work with Neil Robertson and R had completed the proof of the Strong Perfect Graph Con four joint authors presented their work at a workshop held 30 to November 3, 2002 at the American Institute of Mathem Alto, California. The preliminary version of their paper, the workshop, consists of 148 pages. Read all about it chvatal/perfect/spgt.html 0-14