bibkey: elzein2026local authors: Ayman El Zein, Maidoun Mortada year: 2026 title: “Impact of local girth on the S-packing coloring of k-saturated subcubic graphs” doi: 10.48550/arXiv.2603.25113 url: https://arxiv.org/abs/2603.25113v1 claim: “Conjecture 3 states that every 2-saturated subcubic graph is (1,1,2)-packing colorable.” strata_touched:

• D5/S3/Combinatorics/Graph/ElZeinMortadaSaturatedPackingRefutation license: citation-only triage: anchor

El Zein–Mortada saturated subcubic packing conjecture

Verified locator

DOI: 10.48550/arXiv.2603.25113

URL: https://arxiv.org/abs/2603.25113v1

Version: arXiv:2603.25113v1, submitted 2026-03-26. The title and authors match the arXiv record. No journal DOI is asserted.

Statement and definitions

Printed page 27 states:

Moreover, as we did not find a 2-saturated subcubic graph G such that g3(G) >= 4 that is not (1, 1, 2)-packing colorable, we conjecture the following. Conjecture 3 Every 2-saturated subcubic graph is (1, 1, 2)-packing colorable.

Printed page 2 defines a subcubic graph by maximum degree at most three and defines k-saturation by requiring every degree-three vertex to have at most k degree-three neighbours. It defines an S-packing coloring as a partition into classes whose distinct vertices have graph distance greater than the corresponding entry of S.

The displayed definition of g3 on printed page 3 uses the maximum of the local girths of degree-three vertices, while the following prose calls it the smallest such cycle length. The formal refutation concerns only the printed Conjecture 3, which contains no local-girth hypothesis, and makes no inference about which of those readings was intended.

Counterexample scope

The connected graph with graph6 encoding FhcYG has vertices 0 through 6 and edges 01, 04, 12, 16, 23, 34, 35, 45, and 56. Its degrees are [2,3,2,3,3,3,2]; the degree-three vertices have respectively 0, 2, 2, and 2 degree-three neighbours. The triangle on vertices 3, 4, and 5 forces one of them into the radius-two color class. Every vertex lies within distance two of each triangle vertex, leaving a five-cycle after deletion of whichever one is chosen. The remaining vertices therefore cannot be split into two independent classes.

The four degree-three vertices have local girths (5,3,3,3). This datum is not used by the refutation of the unqualified printed conjecture.

Bounded prior-resolution evidence

The preregistration on issue 8799 records a targeted arXiv and repository search before the proof probe. It found the source at version 1 and a later paper resolving other conjectures from the same article, but no resolution of Conjecture 3 in the searched scope. The current arXiv record still lists only version 1. Google Scholar and exhaustive non-arXiv literature coverage were not verified, so no worldwide priority or exhaustive absence claim is made.

The target is a Tier 1 recent named conjecture. The graph, its structural properties, and its non-colorability are established directly in the formal module; no novelty is claimed for general packing-coloring methods.