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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 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 labelingsFind 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 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 nK2 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/.
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