TAILIEUCHUNG - Báo cáo toán học: "Orthogonal Vector"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Orthogonal Vector. | Orthogonal Vector Coloring Gerald Haynes Catherine Park Department of Mathematics Department of Mathematics Central Michigan University University of Pittsburgh hayne1gs@ cpark7486@ Amanda Schaeffer Jordan Webster Department of Mathematics Department of Mathematics University of Arizona Central Michigan University socks4me@ webst1jd@ Lon H. Mitchell Department of Mathematics Applied Mathematics Virginia Commonwealth University lmitchell2@ Submitted Sep 29 2008 Accepted Mar 26 2010 Published Apr 5 2010 Mathematics Subject Classification 05C15 Abstract A vector coloring of a graph is an assignment of a vector to each vertex where the presence or absence of an edge between two vertices dictates the value of the inner product of the corresponding vectors. In this paper we obtain results on orthogonal vector coloring where adjacent vertices must be assigned orthogonal vectors. We introduce two vector analogues of list coloring along with their chromatic numbers and characterize all graphs that have vector chromatic number two in each case. In this paper we define and explore possible vector-space analogues of the list-chromatic number of a graph. The first section gives basic definitions and terminology related to graphs vector representations and coloring. Section 2 introduces vector coloring and the corresponding definitions of the list-vector and subspace chromatic numbers of a graph and presents some results and related problems. In the final section we characterize all graphs that have chromatic number two in each case. Research supported by National Science Foundation Grant 05-52594 and Central Michigan University THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R55 1 1 Vector Coloring We will assume that the reader is familiar with some of the more common definitions in graph theory and graph coloring. For a general introduction the reader is encouraged to refer to Diestel s book 6 on graph theory or Jensen and .

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