Surface Chromatic Width and Bounded-Facewidth Snarks

Definitions and mathematical statement

Let $S_g$ be the closed orientable surface of genus $g\ge0$. For an embedded graph $G$, edgewidth $\operatorname{ew}(G)$ is the minimum length of a noncontractible cycle, and facewidth $\operatorname{fw}(G)$ is the minimum number of intersections with $G$ of a noncontractible simple closed curve in general position. Use infinity if no such curve exists, as on the sphere. The dual $G^*$ requires the source's cellular-embedding convention. A cubic graph is three-edge-colorable if its edges can be colored with three colors so that edges incident at a vertex have different colors.

The source preserves three nested assertions:

1. For every $g\ge0$, $\chi_w(S_g)=4$.
2. More strongly, for every graph $G$ embedded in $S_g$ with $\operatorname{ew}(G^*)\ge3$, the printed assertion is $\phi_c(G)=4$. The source calls this equivalent to three-edge-colorability of every cubic embedded $G$ with $\operatorname{fw}(G)\ge3$ and reports $\operatorname{fw}(G)=\operatorname{ew}(G^*)$ for its cubic embeddings.
3. For every $g\ge0$ there exists $w$ such that every cubic graph embedded in $S_g$ with $\operatorname{ew}(G)\ge w$ is three-edge-colorable.

The last assertion uses primal edgewidth, not dual edgewidth or facewidth. The original passage's informal phrase "snarks on $S_g$ have bounded facewidth" does not by itself justify replacing that displayed condition by another width.

Here $\phi_c$ conventionally denotes circular flow number: the infimum of $r\ge2$ for which an orientation supports a real flow, conserved at every vertex, with every directed edge value in $[1,r-1]$. Under this convention, three-edge-colorability of a bridgeless cubic graph corresponds to a nowhere-zero four-flow and $\phi_c(G)\le4$, not to equality with four. Smaller flow numbers are possible. The literal equality and asserted equivalence therefore require reconciliation; the sign is not silently changed. The excerpt also omits the admissible graph/connectivity hypotheses, which matter for flow existence.

The definition of the chromatic-width invariant $\chi_w$ appears in the missing source display (3). The original passage describes it through circular chromatic numbers $\chi_c$ of sufficiently wide embeddings and claims the equivalent orientable-surface target \[ \forall g\ \forall\varepsilon>0\ \exists W\ \forall G\hookrightarrow S_g, \quad\operatorname{ew}(G)\ge W\Longrightarrow\phi_c(G)<4+\varepsilon, \] with reduction to cubic graphs by lifting [7]. It cites [2] for $\chi_c(G)\le\chi_{\mathrm{loc}}(G)+\varepsilon$ at sufficiently large width and refers to missing display (4). Definitions and hypotheses for $\chi_{\mathrm{loc}}$ and those equivalences are not supplied.

The original historical discussion reports Fisk triangulations with exactly two odd-degree vertices, adjacent to each other, having $\chi(G)\ge5$ [3], and examples of arbitrarily large edgewidth for each positive genus. It gives only an unpublished $\chi_{\mathrm{loc}}<5$ result for a very special torus subclass, and a conditional near-four circular-coloring consequence of assertion 3; these are not universal proofs. It attributes the stronger conjecture to Grünbaum [4].

The literature assessment checked on 21 August 2026 instead describes surface Pfaffian equivalences and cites a Pfaffian-orientation survey. Its fit to these circular-coloring/flow conjectures is unclear. That mismatch, missing definitions and the equality issue prevent treating its open label as an independently verified status conclusion.

Short English statement

The source asks three all-genus width and coloring conjectures, culminating in three-edge-colorability of sufficiently wide cubic embeddings. Its literal circular-flow equality, omitted graph conventions and mismatched Pfaffian literature assessment must be checked before the asserted equivalences can be accepted.

Sources

• Graph Theory (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/03, pp. 135--183, https://doi.org/10.4171/owr/2005/03. Original passage cites [2] for the local/circular comparison, [7] for lifting, Grünbaum [4], and Steve Fisk [3]; full bibliography entries and displays (3), (4) are not supplied.
• Literature assessment checked 21 August 2026: Norine, "survey of Pfaffian orientations," https://thomas.math.gatech.edu/PAP/pfafsurv.pdf; original report, https://doi.org/10.4171/owr/2005/03. The assessment's Pfaffian focus is not reconciled with the corrected target.
• Local UnsolvedMath registry, permanent ID $30000174$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The clean statement is labelled corrected and verified with all three conjectures retained; the full original discussion supplies the historical qualifications recorded above.