Cones of closed alternating walks and trails

• Published in: Linear algebra and its applications Vol. 423; no. 2; pp. 351 - 365
• Main Authors: Bhattacharya, Amitava, Peled, Uri N., Srinivasan, Murali K.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.06.2007; Elsevier Science
• Subjects: Algebra; Alternating walks and trails; Applied sciences; Colored graphs; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Flows in networks. Combinatorial problems; Graph theory; Linear and multilinear algebra, matrix theory; Mathematics; Operational research and scientific management; Operational research. Management science; Sciences and techniques of general use; Alternating walks and trails; 05C70; 90C27; 90C57; Colored graphs; Cone; Vertex(graph); Graph theory; 05C70; 90C27; 90C57; Combinatorial optimization; Reachability; Polyhedron; Upper bound; Extreme ray; Graph colouring; Colored graphs; Alternating walks and trails; Summation; Mathematical programming
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2007.01.013

Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the alternating cone. The integral (respectively, {0, 1}) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices. The latter problem, called alternating reachability, is solved in a companion paper along with related results.