Balanced Signings and the Chromatic Number of Oriented Matroids
| Authors | |
|---|---|
| Year of publication | 2006 |
| Type | Article in Periodical |
| Magazine / Source | Combin. Prob. Computing |
| MU Faculty or unit | |
| Citation | |
| Web | doi |
| Field | General mathematics |
| Keywords | orientable matroid; chromatic number; wiring diagram; balanced signing |
| Description | We consider the problem of reorienting an oriented matroid so that all its cocircuits are as balanced as possible in ratio. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit $B = \{B^+, B^-\}$ satisfies $B^+, B^- \neq \emptyset$. We show that, for some reorientation, every signed cocircuit satisfies \[1/f(r) \leq |B^+|/|B^-| \leq f(r)\], where $f(r) \leq 14\,r^2\ln(r)$, and $r$ is the rank of the oriented matroid. In geometry, this problem corresponds to bounding the discrepancies (in ratio) that occur among the Radon partitions of a dependent set of vectors. For graphs, this result corresponds to bounding the chromatic number of a connected graph by a function of its Betti number (corank) $|E|-|V|+1$. |
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