# Combinatorics and Optimization Seminar

Hamiltonian Properties of the 2-Block-Intersection Graphs of Twofold Triple Systems
Friday, 31 March 2017 - 2:30 pm to 3:00 pm
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Room number:
B004
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No
Cost to attend:
Free of charge
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SPEAKER: Aras Erzurumluoglu

A <i> balanced incomplete block design </i> BIBD$$(v,k,\lambda)$$
is a combinatorial design $$(V,{\cal B})$$ in which
(i) $$|V|=v$$, (ii) $$|B|=k$$ for each block $$B$$ in $${\cal B}$$, and
(iii) each 2-subset of $$V$$occurs in precisely $$\lambda$$ blocks of $${\cal B}$$. A BIBD$$(v,3,2)$$ is called a <i> twofold triple system </i> and denoted TTS$$(v)$$. <br>

Given a combinatorial design $${\cal D}$$ with block set $${\cal B}$$,
the <i> $$i$$-block-intersection graph </i> ($$i$$-BIG) of $${\cal D}$$ is the graph having $${\cal B}$$ as its vertex set, and two vertices $$B_{1}$$ and $$B_{2}$$ adjacent if and only if $$|B_{1} \cap B_{2}| = i$$. <br>

Recent joint work with David Pike settles the spectrum of TTSs
with connected non-hamiltonian 2-BIGs, as well as the spectrum of TTSs with
hamiltonian 2-BIGs (or equivalently, the spectrum of TTSs without and with
cyclic 2-intersecting Gray codes, respectively). In this talk I will present one or more of the techniques that we used to obtain these results.