Let N be any integer. Can we define some graphs with vertex set as the partitions of N? If so, is there any literature is available?
This is a good question.
Integer distance graphs are explored in
J.-H. Kang, Combinatorial number theory,:
http://www.westga.edu/~jkang/CNumTh.pdf
Taking this a step further, consider
D.R. Wood, Acyclic, star and oriented colourings of graph subdivisions, Discrete Math. and Theor. Comp. Sci. 7, 2005, 37-50:
http://www.emis.de/journals/DMTCS/pdfpapers/dm070104.pdf
See Theorem 1, p. 39, which gives a condition for a d-degenerate graph with a vertex partition based on non-negative integers. There are 40 places in this article where vertex partitions are discussed.
This is a good question.
Integer distance graphs are explored in
J.-H. Kang, Combinatorial number theory,:
http://www.westga.edu/~jkang/CNumTh.pdf
Taking this a step further, consider
D.R. Wood, Acyclic, star and oriented colourings of graph subdivisions, Discrete Math. and Theor. Comp. Sci. 7, 2005, 37-50:
http://www.emis.de/journals/DMTCS/pdfpapers/dm070104.pdf
See Theorem 1, p. 39, which gives a condition for a d-degenerate graph with a vertex partition based on non-negative integers. There are 40 places in this article where vertex partitions are discussed.
Two trees on integer partitions
http://11011110.livejournal.com/5126.html
Another partial cube of partitions
http://11011110.livejournal.com/6130.html
What is the dimension of the set of partitions?
http://11011110.livejournal.com/6402.html
Hyper graph is the exampe for grouping the vertices. Similar vertices can be grouped and the edges are between the groups.
Let P,Q be two partitions of n. Then P is called a refinement of Q if it can be obtained by partitioning some parts of Q. The set of all partitions of n is partially ordered by the refinement relation. Now we can consider either the comparability graph or the Hasse diagram of this poset. Thus, P and Q are adjacent in the comparability graph if (say) P is a refinement of Q, and they are adjacent in the Hasse diagram if, moreover, there is no partition lying properly between them.
Please consult this link related to ordered partitions and lattice.
http://math.stackexchange.com/questions/395069/set-partition-refinement-lattice-maximal-chain-length
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Graphs