Enumeration of Colored Tilings on Graphs via Generating Functions
Jos\'e L. Ram\'irez, Diego Villamizar

TL;DR
This paper develops a generating function approach to count and analyze colored tilings on graphs, especially Cartesian products of graphs with paths, generalizing grid tiling enumeration with applications in combinatorics.
Contribution
It introduces a novel combinatorial method using bivariate generating functions to enumerate colored graph partitions, extending tiling enumeration to broader graph families.
Findings
Derived formulas for expected number of blocks in colored graph partitions
Generalized grid tiling enumeration to graph Cartesian products
Provided combinatorial techniques applicable to various graph classes
Abstract
In this paper, we study the problem of partitioning a graph into connected and colored components called blocks. Using bivariate generating functions and combinatorial techniques, we determine the expected number of blocks when the vertices of a graph , for in certain families of graphs, are colored uniformly and independently. Special emphasis is placed on graphs of the form , where is the path graph on vertices. This case serves as a generalization of the problem of enumerating the number of tilings of an grid using colored polyominoes.
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Taxonomy
TopicsAdvanced Graph Theory Research · Cellular Automata and Applications · Advanced Combinatorial Mathematics
