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## Mathematics (General)

Room Chesapeake 8

ap063 W-GRAPHS OVER NON-COMMUTATIVE ALGEBRAS

- Alexander Diaz-Lopez ;
- Matthew Dyer ;

n/a

W-GRAPHS OVER NON-COMMUTATIVE ALGEBRAS

__Alexander Diaz-Lopez__, Matthew Dyer.

*University of Notre Dame, Notre Dame, IN*.

Given a Coxeter system (W, S), a W-graph is a graph, together with additional information that encode a representation (denoted τ-representation) of the Hecke algebra associated to W. We generalize this work by defining W-graphs over non-commutative algebras, which give rise to new representations of Hecke algebras. Various examples are discussed that give rise to several representations of Hecke algebras on quotients of path algebras (over suitable quivers). We discuss the relationship between these representations and the τ-representations. The most interesting example comes from a quotient path algebra that is isomorphic to an ideal of Lusztig's asymptotic Hecke algebra (when defined). This work suggests that the conjecture regarding the existence of the asymptotic Hecke algebra for all Coxeter groups is true.

ap064 A SPECTRUM OF EDGE-COLORING, SUBCUBIC, PLANAR GRAPHS

- Michael Santana ;
- Alexandr Kostochka ;

n/a

A SPECTRUM OF EDGE-COLORING, SUBCUBIC, PLANAR GRAPHS

__Michael Santana__, Alexandr Kostochka.

*University of Illinois at Urbana-Champaign, Urbana, IL*.

In 2009, Muthu, Narayanan, and Subramanian introduced a new generalization of edge-coloring known as k-intersection edge-coloring. This variant requires that an edge-coloring be proper and assigns to every vertex a set containing the colors of its incident edges. We then require that sets corresponding to adjacent vertices have at most k elements in common. When k is at least the maximum degree of a given graph, a k-intersection edge-coloring is equivalent to a proper edge-coloring, and when k = 1, it corresponds to a strong edge-coloring (introduced by Foquet and Jolivet in 1983). Thus, as k ranges from 1 to the maximum degree, this new edge-coloring variant considers a wide spectrum of edge-coloring parameters. The k-intersection chromatic index is the minimum number of colors in a k-intersection edge-coloring of a given graph. In this talk, we restrict our attention to subcubic, planar graphs (i.e., planar graphs with maximum degree 3). As mentioned, when k = 3, the notion of 3-intersection edge-coloring is equivalent to proper edge-coloring. Thus, the 3-intersection chromatic index of subcubic, planar graphs is at most 4 by Vizing's Theorem, and this is known to be best possible. Recently, Kosotochka et al., showed that the 1-intersection chromatic index of subcubic, planar graphs is at most 9. This solves a conjecture of Faudree et al., and is also best possible. We resolve the final case for subcubic, planar graphs by showing that the 2-intersection chromatic index of such graphs is at most 5 and, furthermore, this is best possible.

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Conference Grant Funders

- National Institute of General Medical Sciences – National Institute of Health
- National Security Agency (NSA) - Department of Defense

Platinum Sponsors

- Central Intelligence Agency
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Silver Sponsors

Bronze Sponsors

- Genentech
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- Texas A&M University - Graduate and Professional Studies
- University of Southern California - Keck School of Medicine, Department of Preventive Medicine

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- American Chemical Society
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- Botanical Society of America
- Genetics Society of America
- Maryland Sea Grant
- Society for Industrial and Applied Mathematics
- St. Jude Children’s Research Hospital
- The University of Kansas-Department of Geology