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226. Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groupsŠtefko Miklavič, Primož Šparl, 2012, original scientific article Abstract: In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph ▫$\Gamma$▫ is ▫$n$▫-HC-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton cycle of ▫$\Gamma$▫. Similarly, ▫$\Gamma$▫ is weakly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton path of ▫$\Gamma$▫. Moreover, ▫$\Gamma$▫ is strongly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if for every such path $P$ there is a Hamilton path of ▫$\Gamma$▫ starting with ▫$P$▫. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2-HC-extendable and a complete classification of 3-HC-extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4-HP-extendable.
Keywords: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian group
Published in RUP: 15.10.2013; Views: 2833; Downloads: 143
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227. Hamilton cycles in (2, odd, 3)-Cayley graphsHenry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific article Abstract: In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.
Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group
Published in RUP: 15.10.2013; Views: 3023; Downloads: 133
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228. A spectral proof of the uniqueness of a strongly regular graph with parameters (81, 20, 1, 6)Dragan Stevanović, Marko Milošević, 2009, original scientific article Abstract: We give a new proof that there exists a unique strongly regular graph with parameters (81, 20, 1, 6). Unlike the finite geometry approach used by Brouwerand haemers, we use linear algebra and spectral graph theory concepts, namely the technique of star complements, in our proof.
Keywords: graph theory
Published in RUP: 15.10.2013; Views: 2757; Downloads: 32
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230. Adjacency preservers, symmetric matrices, and coresMarko Orel, 2012, original scientific article Abstract: It is shown that the graph ▫$\Gamma_n$▫ that has the set of all ▫$n \times n$▫ symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if ▫$n \ge 3$▫. Eigenvalues of the graph ▫$\Gamma_n$▫ are calculated as well.
Keywords: adjacency preserver, symmetric matrix, finite field, eigenvalue of a graph, coloring, quadratic form
Published in RUP: 15.10.2013; Views: 3163; Downloads: 141
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