@MISC{Grinbergs_threeconnectedgraphs, author = {Emanuels Grinbergs and Dainis Zeps}, title = {Threeconnected graphs with only one Hamiltonian circuit 1}, year = {} }

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Abstract

We will call graph 1-H-graph if it is threeconnected and it has only one Hamiltonian circuit (Hcircuit). We will say that in the graph G three distinct vertices x, y, z in the given order comprise special triplet – shorter, s-triplet {x, y, z} if 1) there is only one Hamiltonian chain (H-chain) [x...y] with end vertices x, y; 2) there isn’t H-chain [x...z]; 3) and either 3.1) G is threeconnected; or 3.2) G is not threeconnected, but it becomes threeconnected if vertex t and edges tx, ty, tz are added. H-chains [y...z] can be of arbitrary number, or be not at all. Graph G satisfying these conditions will be called preparation. If graphs G and G ’ without common elements have correspondingly s-triplets {x, y, z} and {x’,y’, z’}, then the linking these graphs by edges xy’, yx’, zz ’ will give new graph G’ ’ that is 1-H-graph. Rightly, because of condition 3 G’ ’ is threeconnected. The only H-circuit of G’ ’ is composed from elements [x...y], yx’, [x’...y’], y’x. Indeed, each H-circuit of G’ ’ has just two edges from xy’, yx’, zz’. Because of the condition 1 first two edges go only into indicated H-circuit. Because of the fact that there aren’t H-chains [x...z] in G and [x’...z’] in G’, pairs of edges xy’, zz ’ and yx’, zz ’ do not go in any H-circuit of G’’. 1 This article is compiled from several fragments from Grinbergs manuscripts by D. ZepsIf G is a graph with only one H-circuit we will say that the edges of the H-circuit are strong, but other edges are weak. For each vertex x of G with degree p 3 there are at least 2(p-2) triplets x, y, z that satisfy condition 1 and 2 (Fig. 1, where strong edges are bold). y y’