453 KB41 KB257 KB3 KB2025Alimirzaei, ShirinWitte Morris, Daveštevilka:1, article p1.04letnik:8str. 1-18DOI:10.26493/2590-9770.1795.a62ISSN:2590-9770COBISSID_HOST:285546755URN:URN:NBN:SI:doc-GGOCYVJQenFakulteta za matematiko, naravoslovje in informacijske tehnologijeThe art of discrete and applied mathematicsautomorphismavtomorfizemCayley graphCayleyjev grafcolour-permutingcomplete graphspermutiranje barvpolni grafColour-permuting automorphisms of complete Cayley graphs|Let G be a (finite or infinite) group, and let KG = Cay(G; G \ {1}) be the complete graph with vertex set G, considered as a Cayley graph of G. Being a Cayley graph, it has a natural edge-colouring by sets of the form {s, s-1} for s in G. We prove that every colour-permuting automorphism of KG is an affine map, unless G is isomoprhic to the direct product of Q8 and B, where Q8 is the quaternion group of order 8, and B is an abelian group, such that b2 is trivial for all b in B. We also prove (without any restriction on G) that every colour-permuting automorphism of KG is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D. P. Byrne, M. J. Donner, and T. Q. Sibley in 2013Naj bo G(končna ali neskončna) grupa in naj bo KG = Cay (G; G\{1}) poln graf z množico točk grafa G, obravnavan kot Cayleyjev graf grupe G. Kot Cayleyjev graf ima naravno barvanje povezav z množicami oblike {s, s-1} za s?G\{1}. Dokažemo, da je vsak avtomiorfizem, ki permutira barve grafa KG, afina preslikava, razen če je G~=Q8×B, kjer je Q8 kvaternionska grupa reda 8,B pa abelska grupa, v kateri je element b2 trivialenza vseb?B. Dokažemo tudi (brez kakršnihkoli omejitev v zvezi z G), da je vsak avtomorfizem, ki ohranja barve grafa KG, kompozitum avtomorfizma grupe in avtomorfizma grafa, ki ohranja barve grafa. To je potrditev domneve iz leta 2013, ki so jo postavili D. P. Byrne, M. J. Donner in T. Q. SibleyTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije