
Open problemsDuring 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 decompositionsFind a decomposition of K_{k}[mK_{1}] into C_{r} if r does not divide the number of edges of K_{k} nor does r divide m (and necessary condition that m divides 1/2*k(k1)m^{2} holds). Vertex magic total labelingsFind a VMT labeling of C_{2m} × C_{2n}. Find a VMT labeling of any 2regular graph (other than kC_{n}). Find a VMT labeling of any 3regular graph (other than described by D. McQuillan: every 3regular graph G with a perfect matching F, such that GF consists of two 2regular graphs, each on n vertices). Are there other general methods besides decompositions and Kotzig arrays for finding magictype labelings of large graphs using magictype 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 3regular graph (or at least 3regular 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 cyclesFind 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 nK_{2} for n even (if exists). For more information see homel.vsb.cz/~kov16/publications/graphs2005.php and homel.vsb.cz/~kov16/publications/iwogl2005.php. Antimagic squaresGive 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/.
