A sufficient condition guaranteeing large cycles in graphs

AbstractWe generalize Bedrossian-Chen-Schelp's condition (1993) for the existence of large cycles in graphs, and give infinitely many examples of graphs which fulfill the new condition for hamiltonicity, while the related condition by Bedrossian, Chen, and Schelp is not fulfilled

Symmetry as a sufficient condition for a finite flex

We show that if the joints of a bar and joint framework $(G,p)$ are positioned as `generically' as possible subject to given symmetry constraints and $(G,p)$ possesses a `fully-symmetric' infinitesimal flex (i.e., the velocity vectors of the infinitesimal flex remain unaltered under all symmetry operations of $(G,p)$), then $(G,p)$ also possesses a finite flex which preserves the symmetry of $(G,p)$ throughout the path. This and other related results are obtained by symmetrizing techniques described by L. Asimov and B. Roth in their paper `The Rigidity Of Graphs' from 1978 and by using the fact that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form by means of group representation theory. The finite flexes that can be detected with these symmetry-based methods can in general not be found with the analogous non-symmetric methods.Comment: 26 pages, 10 figure

A generalization of heterochromatic graphs

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, using this criterion, we show that a $g$-chromatic graph $G$ of order $n$ with $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.Comment: 14 pages, 4 figure