Curriculum vitae
Teaching (czech)

Open problems

During my research I found a couple of problems which I believe are interesting. I post some of them here. Some of the problems we are/were working on on the DiMaS seminar.

Please, let me know if you have results, partial results or some other information on any of the problems.

Multipartite graph decompositions

Find a decomposition of Kk[mK1] into Cr if r does not divide the number of edges of Kk nor does r divide m (and necessary condition that m divides 1/2*k(k-1)m2 holds).

Vertex magic total labelings

Find a VMT labeling of C2m × C2n.

Find a VMT labeling of any 2-regular graph (other than kCn).

Find a VMT labeling of any 3-regular graph (other than described by D. McQuillan: every 3-regular graph G with a perfect matching F, such that G-F consists of two 2-regular graphs, each on n vertices).

Are there other general methods besides decompositions and Kotzig arrays for finding magic-type labelings of large graphs using magic-type labelings of small graphs?

Find an infinite class of graphs with an isolate which allows a VMT labeling.

Find a VMT labeling of the Möbious ladder.

Find a VMT labeling with consecutive vertex labels for any 3-regular graph (or at least 3-regular graphs with perfect matching). The condition |G|=0(mod4) must hold.

For more information see www1.cs.columbia.edu/~sanders/graphtheory/people/, homel.vsb.cz/~kov16/publications/ms_thesis.php, and homel.vsb.cz/~kov16/publications/phd_thesis.php.

Vertex antimagic total labelings of cycles

Find a general construction of antimagic labelings with difference 4 and 5.

For more information see www.tuke.sk/baca/publications.htm or homel.vsb.cz/~kov16/publications/ms_thesis.php.

Vertex antimagic total labelings of regular graphs

Find a VAMT labeling of nK2 for n even (if exists).
Or at least find a VAMT labeling of any 3-regular graph.

For more information see homel.vsb.cz/~kov16/publications/graphs2005.php and homel.vsb.cz/~kov16/publications/iwogl2005.php.

Antimagic squares

Give a simple argument, that there is no antimagic square of order 3.

For definitions see mathworld.wolfram.com/AntimagicSquare.html or io.uwinnipeg.ca/~vlinek/jcormie/, for more information see www.geocities.com/~harveyh/anti_ms.htm and for more links see mathworld.wolfram.com/MagicSquare.html or www.geocities.com/~harveyh/.

phone ++420 / 597 325 972
Last update: 29.12.2011