# On a conjecture that strengthens the $k$-factor case of Kundu's $k$-factor Theorem, , a seminar in the Department of Mathematical and Statistical Sciences

Monday, January 31, 2022 11am to 12pm

1201 Larimer Street

Dept. of Mathematical and Statistical Sciences Fall 2021 Seminar Series Presents

Dr. James Shook

Security Components and Mechanisms Group, NIST

WHEN: Monday January 31st, 2022, from 11:00am to noon

TITLE: On a conjecture that strengthens the $k$-factor case of Kundu's $k$-factor Theorem

ABSTRACT: An non-negative sequence of integers $\pi=(d_{1},\ldots,d_{n})$ is said to be
graphic if there exists a graph $G=(V,E)$, called a realization of $\pi$, with
$V=\{v_{1},\ldots, v_{n}\}$  such that $v_{i}\in V$ has $d_{i}$ neighbors in
$G$. In 1974, Kundu showed that for even $n$ if $\pi=(d_{1},\ldots,d_{n})$ is a
non-increasing graphic sequence such that $(d_{1}-k,\ldots,d_{n}-k)$ is graphic,
then some realization of $\pi$ has a $k$-factor. In 1978, Brualdi and then Busch
et al.\, in 2012, conjectured that not only is there a $k$-factor, but there is
$k$-factor that can be partitioned into $k$ edge-disjoint $1$-factors. We will
discuss this conjecture and present some new supporting results.

1 person is interested in this event

Everyone is welcome to join the seminar. Seminar will be hydrid: in-person and virtual (zoom). The speaker will be remote. For in-person, please join us in Student Commons Building room #4017. For virtual attendance, please contact mathstaff@ucdenver.edu for Zoom information.

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