Universal graph
In mathematics, a universal graph is an infinite graph that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado[1][2] and is now called the Rado graph or random graph. More recent work[3] [4] has focused on universal graphs for a graph family Template:Mvar: that is, an infinite graph belonging to Template:Mvar that contains all finite graphs in Template:Mvar. For instance, the Henson graphs are universal in this sense for the Template:Mvar-clique-free graphs.
A universal graph for a family Template:Mvar of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in Template:Mvar; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph[5] so a hypercube can be said to be a universal graph for trees. However it is not the smallest such graph: it is known that there is a universal graph for Template:Mvar-vertex trees, with only Template:Mvar vertices and O(n log n)Script error: No such module "Check for unknown parameters". edges, and that this is optimal.[6] A construction based on the planar separator theorem can be used to show that Template:Mvar-vertex planar graphs have universal graphs with O(n3/2)Script error: No such module "Check for unknown parameters". edges, and that bounded-degree planar graphs have universal graphs with O(n log n)Script error: No such module "Check for unknown parameters". edges.[7][8][9] It is also possible to construct universal graphs for planar graphs that have n1+o(1)Script error: No such module "Check for unknown parameters". vertices.[10] Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with 2n − 2Script error: No such module "Check for unknown parameters". vertices contains every polytree with Template:Mvar vertices as a subgraph.[11]
A family Template:Mvar of graphs has a universal graph of polynomial size, containing every Template:Mvar-vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by O(log n)Script error: No such module "Check for unknown parameters".-bit bitstrings such that an algorithm can determine whether two vertices are adjacent by examining their labels. For, if a universal graph of this type exists, the vertices of any graph in Template:Mvar may be labeled by the identities of the corresponding vertices in the universal graph, and conversely if a labeling scheme exists then a universal graph may be constructed having a vertex for every possible label.[12]
In older mathematical terminology, the phrase "universal graph" was sometimes used to denote a complete graph.
The notion of universal graph has been adapted and used for solving mean payoff games.[13]
References
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- ↑ Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17.
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External links
- The panarborial formula, "Theorem of the Day" concerning universal graphs for trees