The odd independence number of graphs, I: Foundations and classical classes

TL;DR

This work introduces the odd independence number $α_{od}(G)$ and the strong odd chromatic number $χ_{so}(G)$ as complementary graph parameters to classical independence and chromatic notions, establishing the fundamental inequality $α_{od}(G)·χ_{so}(G) ≥ |G|$ and comparing to the classical bound $α(G)χ(G)≥|V|$. It proves that for claw-free graphs one has $α_{od}(G)=α(G^2)$ and $χ_{so}(G)=χ(G^2)$, and it shows NP-hardness of computing $α_{od}(G)$, even on line graphs, via reductions from Maximum Induced Matching. The paper develops general inequalities linking $α_{od}$, $χ_{so}$, and basic parameters, and it derives lower bounds on $α_{od}$ from known $χ_{so}$ bounds across numerous graph families, including paths, cycles, Moore graphs, Kneser graphs, and several graph products and subdivisions, with substantial attention to hypercubes and complements of triangle-free graphs. It further introduces recursive constructions using automorphisms to obtain large $α_{od}$ and analyzes how these ideas apply to hypercubes and to diameter-related properties of triangle-free complements, concluding with a rich set of open problems and directions for part II. The results illuminate how odd independence interacts with structural graph classes and computational complexity, and they provide a suite of exact or tight bounds for a broad spectrum of classical graphs and constructions, guiding future study in the area.

Abstract

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $α_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $χ_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $α_{od}(G)\cdotχ_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. We develop several basic inequalities concerning $α_{od}(G)$, and use already existing results on strong odd coloring, to derive lower bounds for odd independence in many families of graphs. We prove that $α_{od}(G) = α(G^2)$ holds for all claw-free graphs $G$, and apply this result to prove that determining $α_{od}(G)$ is in general NP-hard (and also when restricted to line graphs). We also present many results, using various techniques, concerning the odd independence number of cycles, paths, Moore graphs, Kneser graphs, the complete subdivision $S(K_n)$ of $K_n$, the half graphs $H_{n,n}$, and $K_p \Box K_q$. Further, we consider the odd independence number of the hypercube $Q_d$ and also of the complements of triangle-free graphs. Many open problems for future research are stated. Further related results can be found in part II of this work, arXiv: 2510.01897.

Figures (1)

• Figure 1: Two 3-regular graphs of order 8

Theorems & Definitions (59)

• Proposition 1
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• Proposition 2
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• Proposition 5
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