## g_Z~i[

### Z~i[ uL^(2016N4112017N130)

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 2017N0123()16:30-- cmwLpXnz6K14-631B Carol Zamfirescu (Ghent University, Belgium) Hamiltonian Properties of Polyhedra In this talk we will discuss several related problems concerning hamiltonian properties of planar 3-connected graphs, which we will call polyhedra. The presentation will be divided into four parts, with emphasis on new results, the techniques behind them, and open research problems. The first topic deals with 3-(vertex-)cuts in polyhedra, where we present a strengthening of the classic theorem of Tutte that 4-connected polyhedra are hamiltonian (joint work with Gunnar Brinkmann). This result also strengthens a theorem of Jackson and Yu on triangulations. In order to provide context, we end this first part by giving an overview of results on hamiltonian properties of polyhedra with few 3-cuts (based on joint work with Kenta Ozeki and Nico Van Cleemput). In the second part of the talk, we present a geometrically motivated generalisation of Halin graphs, and study the hamiltonian properties of this family of polyhedra. We strengthen a result of Bondy. This is based on joint work with Boris Schauerte and Tudor Zamfirescu. The third topic concerns new bounds for the orders of the smallest $k$-regular polyhedra, $k \in \{3,4,5\}$, which are non-hamiltonian or non-traceable. We improve bounds of Zaks and Owens. This is joint work with Nico Van Cleemput. The talk ends with a study of 3-fragments (and in consequence cubic vertices) in planar hypohamiltonian graphs (which are necessarily 3-connected, and thus polyhedra). If time permits, more details -- such as sketches of proofs -- will be given for each of the above topics.
 2017N0116()16:30-- cmwLpXnz6K14-631B iwCJSTCERATOC͌ыOtvWFNgj Signature of 3-edge-colorings of cubic graphs
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 2016N1205()16:30-- cmwLpXnz6K14-631B |{Fqicwj Grabbing game on C4-unicyclic graphs
 2016N1121()16:30-- cmwLpXnz6K14-631B Ê_ j On a spanning tree with specified leaves in bipartite graphs
 2016N1114()16:30-- cmwLpXnz6K14-631B iwCJSTCERATOC͌ыOtvWFNgj k-edge-connected factors with degree constraints
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 2016N1031()16:30-- cmwLpXnz6K14-631B icmwj On a properly colored 2-factor II
 2016N1024()16:30-- cmwLpXnz6K14-631B iwCJSTCERATOC͌ыOtvWFNgj Rainbow K1,3 rainbow P4+ ̂ȂӒF
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 2016N1003()16:30-- cmwLpXnz6K14-631B icmwj On a properly colored 2-factor
 2016N0926()16:30-- cmwLpXnz6K14-631B y ĈiCwj HIST in fullerene graph
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 2016N0711()16:30-- cmwLpXnz6K14-631A iwCJSTCERATOC͌ыOtvWFNgj 3ʃOt̕ II
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 2016N0627()16:30-- cmwLpXnz6K14-631A iwCJSTCERATOC͌ыOtvWFNgj 3ʃOt̕
 2016N0620()16:30-- cmwLpXnz6K14-631A icmwj The existence of k-forests and its applications II
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 2016N0530()16:30-- cmwLpXnz6K14-631A Carol Zamfirescu (Ghent University, Belgium) Hypohamiltonian Graphs A graph is hypohamiltonian if it is non-hamiltonian but, when omitting an arbitrary vertex, it becomes hamiltonian. A problem of Sousselier from 1963 initiated the study of these graphs. The smallest hypohamiltonian graph is the famous Petersen graph on 10 vertices. Chvátal asked in 1972 whether there exist planar hypohamiltonian graphs, while Grünbaum conjectured that these graphs do not exist. An infinite family of such graphs was subsequently found by Thomassen, the smallest among them having order 105. We present the progress made towards finding the smallest planar hypohamiltonian graphs, while pointing out the importance of Grinberg's Criterion and discussing related problems on longest paths and longest cycles. We will also treat the planar cubic case, girth restrictions and hypohamiltonian graphs in the context of crossing numbers. A theorem concerning the number of cubic vertices in a planar hypohamiltonian graph---where we use a recent result obtained together with Gunnar Brinkmann and strengthen a theorem of Thomassen---will also be presented. In the second part, we look at almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. We study Gallai's problem on longest paths and its connection with almost hypotraceable graphs---combining our techniques with computational results of McKay, we shall determine the smallest 3-connected cubic almost hypotraceable graph. If time permits, we shall also discuss non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. Connections will be drawn with recent work of Wiener, and we will present solutions to certain problems raised by him related to the minimum leaf number.
 2016N0516()16:30-- cmwLpXnz6K14-631A icmwj The existence of k-forests and its applications
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 2016N0418()16:30-- cmwLpXnz6K14-631A iwCJSTCERATOC͌ыOtvWFNgj Proper k-connection of bipartite graphs
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