Characterizing Flag Graphs and Induced Subgraphs of Cartesian Product Graphs

Abstract

The vertices of the flag graph (P ) of a graded poset P are its maximal chains. Two vertices are adjacent whenever two maximal chains differ in exactly one element. In this paper we characterize induced subgraphs of Cartesian product graphs and flag graphs of graded posets. The latter class of graphs lies between isometric and induced subgraphs of Cartesian products in the embedding structure theory. Both characterization use certain edge-labelings of graphs. Mathematics Subject Classifications (2000):

Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle

“…We note that with a similar method induced subgraphs of Hamming graphs in particular [17] and induced subgraphs of Cartesian graphs in general [23] can be treated.…”

“…There are many classes of graphs that are naturally defined as (metric) subgraphs of Cartesian products, see [1], [4], [6], [17], [26] for a sample of such references. Graphs that are subgraphs of general Cartesian products has been studied as well, see [2], [16], [19], [23] where several characterizations of these graphs are proved.…”

Realizability of Polytopes as a Low Rank Matrix Completion Problem

“…Many other related constructions can be found in [5], which presents combinatorial operations on a hierarchy of such objects, but here we are only interested in the minimal properties we need for Theorem 1. These are really properties of a poset's flag graph, which have been characterized by Peterin [6].…”