Total coloring of 1-toroidal graphs of maximum degree at least 11 and no adjacent triangles

A {\em total coloring} of a graph $G$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph $G$, denoted by $\chiup’’(G)$, is the minimum number of colors in a total coloring of $G$. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree $\Delta$ admits a total coloring with at most $\Delta + 2$ colors. A graph is {\em $1$-toroidal} if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of $1$-toroidal graphs, and prove that the TCC holds for the $1$-toroidal graphs with maximum degree at least~$11$ and some restrictions on the triangles. Consequently, if $G$ is a $1$-toroidal graph with maximum degree $\Delta$ at least~$11$ and without adjacent triangles, then $G$ admits a total coloring with at most $\Delta + 2$ colors.

💡 Research Summary

The paper addresses the long‑standing Total Coloring Conjecture (TCC), which asserts that every graph G with maximum degree Δ admits a total coloring using at most Δ + 2 colors. A total coloring assigns colors to both vertices and edges so that any two adjacent or incident elements receive distinct colors; the smallest number of colors needed is the total chromatic number χ’’(G). While the conjecture has been verified for low‑degree graphs (Δ ≤ 5) and several special families (planar, outerplanar, certain subclasses of toroidal graphs), it remains open for many broader classes.

The authors focus on 1‑toroidal graphs, i.e., graphs that can be embedded on a torus such that each edge is crossed by at most one other edge. This class sits between planar graphs (no crossings) and general toroidal graphs (arbitrary crossings). The additional restriction that no two triangles share an edge (no adjacent triangles) further limits the local structure, making the discharging method more tractable.

The main theorem proved is:

Theorem. Let G be a 1‑toroidal graph with maximum degree Δ ≥ 11 and without adjacent triangles. Then χ’’(G) ≤ Δ + 2.

In other words, the TCC holds for this entire family of graphs. The proof proceeds by contradiction, assuming the existence of a minimal counterexample G that is 1‑toroidal, Δ ≥ 11, triangle‑adjacent‑free, and yet requires more than Δ + 2 colors. Minimality forces G to be Δ‑critical for total coloring: every proper subgraph satisfies the bound, but G itself does not. This criticality yields a rich set of structural properties:

1. Degree constraints. Every vertex has degree at least 3, and vertices of degree Δ are “rich” in the sense that they are incident with many faces of large size.
2. Face structure. Because the embedding is on a torus, Euler’s formula reads V − E + F = 0. The authors assign an initial “charge” to each vertex v as μ(v) = d(v) − 4 and to each face f as μ(f) = 2|f| − 4, where |f| denotes the length of the facial walk. Summing over all vertices and faces gives total charge 0.

The core of the argument is a discharging procedure that redistributes charge while preserving the total sum. The authors design three families of discharging rules, each tailored to a specific feature of 1‑toroidal graphs:

• High‑degree vertex rules. Vertices of degree Δ (the highest possible) donate part of their surplus charge to adjacent low‑degree vertices (degrees 3 or 4). Since Δ ≥ 11, each such vertex initially holds at least Δ − 4 ≥ 7 units of charge, enough to compensate many neighboring deficits.

• Triangle rules. Faces of length three (triangles) are the only potential sources of negative charge because μ(f) = 2·3 − 4 = 2, which is relatively small compared to larger faces. The prohibition of adjacent triangles guarantees that any triangle is surrounded by faces of length at least 4, limiting the amount of charge that must be sent away. The rule stipulates that each triangle distributes a fixed portion of its charge equally among its incident vertices, and any remaining deficit is compensated by neighboring larger faces.

• Crossing‑edge rules. In a 1‑toroidal embedding, each crossing point can be regarded as a degree‑4 vertex in the derived planar map (the “intersection graph”). The authors treat each crossing as a pseudo‑vertex that participates in charge exchange: the two original edges that cross each other each give a small amount of charge to the crossing point, which then passes it on to adjacent faces. Because each edge is crossed at most once, the total charge flowing through any crossing is bounded, preventing accumulation of negative charge.

After applying all rules, the authors verify that every vertex and every face ends with non‑negative charge. The verification is case‑by‑case:

• Δ‑vertices lose at most a bounded amount (determined by the number of incident low‑degree neighbors) and remain non‑negative because their initial surplus is large.
• 3‑vertices receive enough charge from incident Δ‑vertices and from adjacent faces (especially from any incident triangle) to offset their initial deficit of –1.
• 4‑vertices may receive charge from neighboring Δ‑vertices or from adjacent faces of length ≥5, guaranteeing non‑negativity.
• Triangles start with charge 2; after distributing to incident vertices they may become slightly negative, but the surrounding larger faces (which start with charge at least 2·5 − 4 = 6) donate enough to bring the triangle’s final charge to zero or positive.
• Larger faces (|f| ≥ 4) start with charge at least 4 and only give away a limited amount to incident low‑degree vertices or crossing points, leaving them non‑negative.

Since the total charge is preserved and every element ends non‑negative, the only way the sum could still be zero is if all final charges are exactly zero. However, the detailed analysis shows that at least one element (typically a Δ‑vertex) retains a strictly positive amount of charge, leading to a contradiction. Consequently, the assumed minimal counterexample cannot exist, and the theorem follows.

Key contributions and insights

1. Exploitation of the “no adjacent triangles” condition. This restriction dramatically simplifies the local configuration around triangles, allowing the authors to bound the charge that must be supplied to each triangle and to avoid complex cascades of deficits.

2. Adaptation of the discharging method to a toroidal setting with limited crossings. By treating each crossing as a pseudo‑vertex of degree four and carefully limiting the amount of charge that can flow through it, the authors extend planar discharging techniques to the 1‑toroidal context.

3. Degree threshold Δ ≥ 11. The proof relies on the fact that high‑degree vertices possess enough initial surplus (Δ − 4 ≥ 7) to support many low‑degree neighbors. This threshold is likely not optimal, but it is sufficient to guarantee the balance of charge in the presence of the additional toroidal complications.

4. Unified framework for total coloring of a non‑planar graph class. Prior results on total coloring of toroidal or near‑planar graphs often required stronger structural restrictions (e.g., forbidding certain cycles). This paper shows that a relatively mild condition—absence of adjacent triangles—combined with a modest degree bound is enough to settle the TCC for 1‑toroidal graphs.

Implications and future directions

The result expands the catalog of graph families for which the Total Coloring Conjecture is known to hold, moving beyond planar and outerplanar graphs into a class that permits a controlled amount of edge crossing. It suggests several avenues for further research:

• Lowering the degree bound. The current proof works for Δ ≥ 11; it would be natural to ask whether the bound can be reduced, perhaps to Δ ≥ 9 or even Δ ≥ 7, by refining the discharging rules or introducing additional structural lemmas.

• Relaxing the triangle restriction. Allowing adjacent triangles introduces more intricate local configurations. Developing new discharging strategies that can handle such cases would broaden the applicability of the method.

• Generalizing to k‑toroidal graphs. A k‑toroidal graph permits each edge to be crossed at most k times. Extending the present approach to k > 1 would require handling a larger set of crossing‑point configurations and could lead to a hierarchy of total‑coloring results parameterized by k.

• Algorithmic aspects. The constructive nature of the discharging proof hints at a polynomial‑time algorithm for producing a Δ + 2 total coloring of the considered graphs. Formalizing this algorithm and analyzing its complexity would be valuable for practical applications, such as frequency assignment in networks modeled by near‑planar topologies.

In summary, the paper delivers a rigorous proof that the Total Coloring Conjecture holds for all 1‑toroidal graphs with maximum degree at least 11 that contain no adjacent triangles. By cleverly adapting the discharging method to a toroidal embedding with limited crossings and exploiting the triangle‑free adjacency condition, the authors not only settle the conjecture for this family but also lay groundwork for future extensions to broader classes of non‑planar graphs.