Lexicographic product of graphs

In graph theory, the lexicographic product or graph composition "G" ∙ "H" of graphs "G" and "H" is a graph such that
* the vertex set of "G" ∙ "H" is the cartesian product "V(G)" $imes$ "V(H)"; and
* any two vertices "(u,u')" and "(v,v')" are adjacent in "G" ∙ "H" if and only if either "u" is adjacent with "v", or "u" = "v" and "u' " is adjacent with "v' ".

If the edge relations of the two graphs are order relations, then the edge relation of their lexicographic product is the corresponding lexicographic order.

The lexicographic product was first studied by Felix Hausdorff (1914). As Feigenbaum and Schäffer (1986) showed, the problem of recognizing whether a graph is a lexicographic product is equivalent in complexity to the graph isomorphism problem.

Properties

The lexicographic product is in general noncommutative: "G" ∙ "H" ≠ "H" ∙ "G". However it satisfies a distributive law with respect to disjoint union: ("A" + "B") ∙ "C" = "A" ∙ "C" + "B" ∙ "C".In addition it satisfies an identity with respect to complementation: C("G" ∙ "H") = C("G") ∙ C("H").

The independence number of a lexicographic product may be easily calculated from that of its factors::α("G" ∙ "H") = α("G")α("H")(Geller and Stahl 1975).

The chromatic number of a lexigraphic product is as well multiplicative:

$omega(Gcdot\; H)\; =\; omega(G)cdot\; omega(H)$

Application

In 1972 Abbott gave a curios super-quadratic constructive lower bound for the Ramsey numbers: Suppose you have a "n"-vertex graph "G" with independence number and clique number smaller than "n¹⁰" (or any other power) The same relations hold for the graph "G" ∙ "G" or any larger powers of "G". This gives a infinite sequence of explicit Ramsey graphs provided that we have the graph to start from. By checking all the graphs on k^10 vertices (k large enough), one of them certainly will be "k"-Ramsey.

References

*cite journal
author = Feigenbaum, J.; Schäffer, A. A.
year = 1986
title = Recognizing composite graphs is equivalent to testing graph isomorphism
journal = SIAM J. Comput.
volume = 15
pages = 619–627
id = MathSciNet | id = 0837609
doi = 10.1137/0215045

*cite journal
author = Geller, D.; Stahl, S.
title = The chromatic number and other functions of the lexicographic product
journal = Journal of Combinatorial Theory, Series B
volume = 19
year = 1975
pages = 87–95
id = MathSciNet | id = 0392645
doi = 10.1016/0095-8956(75)90076-3

*cite book
author = Hausdorff, F.
authorlink = Felix Hausdorff
title = Grundzüge der Mengenlehre
year = 1914
location = Leipzig

*cite book
author = Imrich, Wilfried; Klavžar, Sandi
title = Product Graphs: Structure and Recognition
publisher = Wiley
year = 2000
id = ISBN 0-471-37039-8

*cite article
author = Abbott, H. L.
title = Lower bounds for some Ramsey numbers
journal = Discrete Math.
volume = 2
year = 1972

External links

*mathworld | title = Graph Lexicographic Product | urlname = GraphLexicographicProduct