arXiv subjects

Arnar Á. Kristjánsson

Publications and source records attributed to Arnar Á. Kristjánsson.

3 recordsLinked to original sources

arXiv2026

Counterexamples to the Strong Roberson Conjecture

We refute the Strong Roberson Conjecture, which asserts that adding any graph outside a class closed under minors and disjoint unions strictly increases the distinguishing power of homomorphism counts from that class. More precisely, we construct connected graphs $H$ for which counts from graphs excluding $H$ as a minor determine the number of homomorphisms from $H$ to any target graph. We also refute the analogous conjecture with immersions in place of minors. We give explicit infinite families of excluded graphs, including cubic bipartite graphs that yield counterexamples for both relations. The proof introduces a method for deriving exact homomorphism count dependence from modular equivalences. We obtain these equivalences for infinitely many primes using prime-order automorphisms of graphs that exclude their orbit quotients as minors or immersions.

math.CO↗

arXiv2026

Oddomorphisms, Split-Off Minors, and the Strong Roberson Conjecture

We show that the existence of an oddomorphism from a graph $F$ to a graph $G$ does not imply that $G$ is a minor of $F$. This answers a question posed by Roberson (2022) and shows that the CFI graphs cannot be used to prove the Strong Roberson Conjecture. Additionally, we introduce the concept of a split-off minor and show that the existence of an oddomorphism from $F$ to $G$ implies that $G$ is a split-off minor of $F$. Consequently, every class that is closed under taking split-off minors and disjoint unions is homomorphism distinguishing closed. The split-off minor relation is the first minor-like structural relation shown to have this property, marking a meaningful advancement in our understanding of the interaction between structural graph containment and homomorphism indistinguishability relations.

cs.DM↗

arXiv2025

Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts

A query algorithm based on homomorphism counts is a procedure to decide membership for a class of finite relational structures using only homomorphism count queries. A left query algorithm can ask the number of homomorphisms from any structure to the input structure and a right query algorithm can ask the number of homomorphisms from the input structure to any other structure. We systematically compare the expressive power of different types of left or right query algorithms, including non-adaptive query algorithms, adaptive query algorithms that can ask a bounded number of queries, and adaptive query algorithms that can ask an unbounded number of queries. We also consider query algorithms where the homomorphism counting is done over the Boolean semiring $\mathbb{B}$, meaning that only the existence of a homomorphism is recorded, not the precise number of them.

cs.LO↗