Alphabeta MathHow statement and proof provenance work
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Maximal proper modules need not be disjoint when the graph or its complement is disconnected
Statement refuted
In every graph, the maximal proper modules are pairwise disjoint.
Facts & Assumptions
Given: The edgeless graph on vertices .
A set is a module when every outside vertex is complete or anticomplete to it (Modules of a graph, and the trivial modules).
In a connected and anticonnected graph, overlapping proper modules do force a larger proper module and maximal proper modules are disjoint (In a connected and anticonnected graph, the union of two proper modules that meet is again a proper module, In a connected and anticonnected graph with at least two vertices, each vertex lies in a largest proper module, and two such modules are equal or disjoint).
A set is a module of a graph exactly when it is a module of the complement (A vertex set is a module of exactly when it is a module of ).
Counterexample
In the edgeless graph , every subset is a module, because every outside vertex is anticomplete to it.
The sets and are proper modules, and each is maximal among proper modules because the only larger module containing it is the whole vertex set.
These two maximal proper modules meet in , so they are not pairwise disjoint.
By [L3], the same two sets are also overlapping maximal proper modules in the complement graph , which is connected while its complement is disconnected.
Therefore the conclusion of [L2] genuinely needs the connected-and-anticonnected hypotheses.
Depends on
- Modules of a graph, and the trivial modules
- A vertex set is a module of $G$ exactly when it is a module of $\overline G$
- In a connected and anticonnected graph, the union of two proper modules that meet is again a proper module
- In a connected and anticonnected graph with at least two vertices, each vertex lies in a largest proper module, and two such modules are equal or disjoint
- Modular partitions and the quotient graph they define
- Prime graphs: those whose only modules are the trivial ones
- Connected graphs and connected components defined by the existence of vertex paths
- Anticonnected graphs and anticonnected components
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- T. Harju, Lecture Notes on Combinatorial Structures in Graph Theory, Remark 4.1 (standard reference, not scraped)