On a conjecture about the strong odd chromatic number of planar graphs

A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of $v$. The strong odd chromatic number of $G$ is defined as the smallest integer $k$ for which $G$ admits a strong odd coloring using $k$ colors. In this paper, we evaluate the strong odd chromatic number of join of cycles and empty graphs and one point union of graphs. Using these results, we construct infinite family of planar graphs that serves as counter examples to a recent conjecture regarding the upper bound of the strong odd chromatic number of planar graphs.

💡 Research Summary

The paper investigates the strong odd coloring of graphs, a refinement of ordinary proper coloring introduced by Petruševski and Skrekovski (2022) and later strengthened by Kwon and Park (2024). In a strong odd coloring, for every vertex v and every color c, either c appears an odd number of times among the neighbors of v or it does not appear at all. The strong odd chromatic number χₛₒ(G) is the smallest number of colors that admits such a coloring. While it is known that every planar graph is odd‑8‑colorable, the exact bound for χₛₒ on planar graphs has remained open. A recent conjecture (Pang, Miao, and Fan, 2026) claimed that 13 colors always suffice for planar graphs. This work disproves that conjecture by constructing infinitely many planar graphs whose strong odd chromatic number exceeds 13, and by exhibiting a planar graph with χₛₒ = 17, the highest value currently known.

The authors’ approach rests on two graph operations: (1) the join of a cycle Cₙ with an empty graph Kₘ, and (2) the one‑point union (identifying a designated vertex across copies). For the join operation, they first observe that any color used on the cycle part must form an odd color class within the cycle, because the vertices of Kₘ are adjacent to all cycle vertices. Moreover, restricting a strong odd coloring to the cycle yields a 2‑distance coloring, limiting the frequency of each color to at most ⌊n/3⌋. Using these observations, Theorem 1 determines the exact strong odd chromatic number of the wheel graph Wₙ = Cₙ ∨ K₁. The value depends on n modulo 6: χₛₒ(Wₙ) equals 4, 5, 6, 7, or n + 1 for small n, with precise conditions listed in the theorem. The proof combines the frequency bound with parity constraints on color classes, showing that for certain residues a minimum of four colors is needed, while for others the parity forces an odd number of colors, raising the lower bound to five or six.

Theorem 2 extends the result to the general join Cₙ ∨ Kₘ. Since all vertices of Kₘ are mutually adjacent to the whole cycle, the colors used on the cycle cannot be reused on Kₘ. If m is odd, a single color can color all vertices of Kₘ; if m is even, one extra color is required. Consequently χₛₒ(Cₙ ∨ Kₘ) equals χₛₒ(Wₙ) when n is odd, and χₛₒ(Wₙ) + 1 when n is even.

The second operation, the one‑point union, is defined as Iₓ(G₁,…,G_k), where a designated vertex x_i in each graph G_i is identified into a single vertex x. This operation preserves planarity. The authors consider Gₙ = Cₙ ∨ K₂, with the two vertices of K₂ named xₙ and yₙ. They then form I_y(Gₘ, Gₙ) by identifying the y‑vertices of Gₘ and Gₙ. Theorem 3 proves that χₛₒ(I_y(Gₘ, Gₙ)) = χₛₒ(Wₘ) + χₛₒ(Wₙ) – 1. The argument is that the common vertex y is adjacent to the entire vertex sets of both cycles, forcing the color sets used on the two cycles to be disjoint; the identified vertex y itself must receive a new color, while the remaining vertices xₘ and xₙ can reuse colors from the opposite cycle. Hence the total number of colors is the sum of the two wheel chromatic numbers minus one.

By plugging concrete values of m and n into Theorem 3, the authors obtain a whole family of planar graphs with strong odd chromatic numbers ranging from 14 up to 17. For example:

• I_y(G₇, G₁₅) has χₛₒ = 14,
• I_y(G₆, G₈), I_y(G₇, G₇), and I_y(G₈, G₁₅) have χₛₒ = 15,
• I_y(G₇, G₈) has χₛₒ = 16,
• I_y(G₈, G₈) has χₛₒ = 17 (illustrated in Figure 2).

All these graphs are planar, providing infinitely many counterexamples to the conjecture that 13 colors always suffice. Moreover, the construction shows that χₛₒ can be arbitrarily large within the planar class, though the current maximum exhibited is 17. The paper concludes with three open problems: (1) find planar graphs with χₛₒ > 17; (2) improve the known universal upper bound of 388 to a smaller constant; (3) determine a constant c such that only finitely many planar graphs have χₛₒ > c.

In summary, the work delivers a precise analysis of strong odd colorings for joins of cycles with empty graphs and for one‑point unions, and leverages these results to disprove a recent conjecture about planar graphs. The methodology highlights how elementary graph operations can be exploited to control parity constraints in strong odd colorings, opening new avenues for tightening bounds on χₛₒ in planar and more general graph families.