Rainbow Threshold Graphs

TL;DR

This work introduces $k$-rainbow threshold graphs as a colored generalization of threshold graphs and develops a rainbow-sequence framework to encode color-based adjacency. It establishes a color-stratified landscape of graph classes and proves that asymptotically almost surely a $(k+1)$-rainbow graph cannot be isomorphic to any $k$-rainbow graph, via analysis of neighborhood-equivalence classes and the recoverability of color information (AC, ASC, and $\ell$-good notions). It also proves that the collection of $k$-rainbow threshold graphs does not satisfy the $0$-$1$ law for first-order logic for any fixed $k$, by showing a fixed FO sentence has a nontrivial limiting probability. Together, these results illuminate how adding color information yields new graph structures and alters logical limit behaviors, deepening the understanding of generalized threshold graphs and their logical limits.

Abstract

We define a generalization of threshold graphs which we call $k$-rainbow threshold graphs. We show that the collection of $k$-rainbow threshold graphs do not satisfy the $0$-$1$ law for first order logic and that asymptotically almost surely all $(k+1)$-rainbow threshold graphs are not isomorphic to a $k$-rainbow threshold graph.

Theorems & Definitions (46)

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