Multipartite graph decomposition: cycles and closed trails

This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not K_n ), in the case that the cycle lengths are “small”. Cycles up to length n are considered, when the complete multipartite graph has n parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned

Multipartite graph decomposition: cycles and closed trails

This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not <em>K</em>_<em>n</em> ), in the case that the cycle lengths are “small”. Cycles up to length <em>n</em> are considered, when the complete multipartite graph has <em>n</em> parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned.<br /

Purification and Characterisation of the Repressor Proteins from the Streptomyces Bacteriophage OC31

OC31 is a bacteriophage of Streptomyces species in which it undergoes a lysogenic life-cycle. The phage possesses a 41.5kb genome and cohesive ends. The lysogenic phase of development is regulated by the repressor gene (c) which lies in the centre of the phage genome (Lomovskaya et aU 1972). The c gene was cloned on a 3.4kb SphI-G fragment; it was sequenced and mapped at the transcriptional level (Sinclair and Bibb, 1988; Sinclair and Bibb, 1989). Further low resolution SI mapping identified that transcription of the repressor gene consisted of 3 nested N- terminally different, C-terminally identical, in-frame mRNA's (Smith and Owen, appendix). Primer extension analysis mapped the 5' ends of the transcripts; upstream of each was found a good consensus -10 promoter sequence and identically conserved ribosome binding sites situated 7-10bp upstream of the ATG initiation codon. The three transcripts each possessed an ORF predictedto produce proteins of 74, 54 and 42kDa (Smith and Owen, appendix). The repressor gene was expressed on a high copy number plasmid in E.coli and proteins unique to the c gene were analysed. Initial work showed that in E.coli, a promoter located upstream of the c gene, cP1 was functional, but that promoters internal to the c gene, cP2 and cP3 did not. Three overexpressed proteins of 110, 70 and 50kDa were observed on SDS-PAGE gels of crude lysates of E.coli carrying the repressor gene. The introduction of frameshift mutations at the unique Ncol restriction site revealed that the 74kDa repressor actually migrated on SDS-PAGE with the mobility of a protein of 1 10kDa, and that the observed 70 and 50kDa repressor proteins were actually the predicted 54 and 42kDa proteins. The 74,54 and 42kDa repressor proteins were purified away from E.coli proteins. This enabled N-terminal protein sequencing of the 54 and 42kDa proteins which confirmed the identity of the predicted ORF assigned for each protein. A plasmid, pMS221, encoding only the 42kDa protein was used to isolate the 42kDa protein in isolation. Ammonium sulphate precipitation followed by anion-exchange and heparin-agarose chromatography resulted in 95% pure protein. Gel filtration and chemical crosslinking identified that the pure 42kDa repressor most probably exists as a tetramer. DNA binding studies using the bandshift assay demonstrated that the 42kDa repressor binds to all three putative promoters in the repressor gene itself with varying degrees of strength. It appears to bind the cP2 promoter with the greatest affinity followed by the cP1 promoter and finally cP3. The cP1 and cP2 promoters contain a 20bp repeated sequence and is a putative operator sequence. DNasel footprinting of the 42kDa protein binding to the cP1 promoter was attempted. The 42kDa repressor also binds to a site within the early region of the phage. This site possesses two putative operator sites interspaced by a terminator-like sequence. Each half of this site, containing only a single 20mer motif was shown to bind the 42kDa protein with relatively low affinity. However, when the entire site was used DNA-binding was found to be very strong and comparable to that of the cP2 promoter

Packing lambda-fold complete multipartite graphs with 4-cycles

A maximum packing of any lambda-fold complete multipartite graph (where there are lambda edges between any two vertices in different parts) with edge-disjoint 4- cycles is obtained and the size of each minimum leave is given. Moreover, when lambda =2, maximum 4-cycle packings are found for all possible leaves

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum