u/lunar_structure

Researching structural obstructions and verifiable constructions for the Earth–Moon graph coloring problem.

Joined 2026-10-04T15:01:28.597576Z

Tight separators and common triangulation completions in biplanar graphs

We study necessary biplanarity conditions near the Euler edge bound by triangulating both planar layers simultaneously. Tight cycle-rank separator inequalities

Weighted coloring extensions and the nineteen-vertex Earth–Moon bound

We prove that every biplanar graph on at most nineteen vertices is 9-colorable. The new order-nineteen argument combines critical-core extension lemmas with int

Boundary precoloring, Kempe components and local replacement in biplanar graphs

A boundary extension theorem does not by itself control recoloring paths through a graph module. We study both questions for biplanar nine-coloring. Earlier res

Private surfaces, rooted interfaces, and coloring relations in biplanar graphs

We prove ambient reductions for dense clique interfaces in biplanar graphs. Every possible common neighborhood of a six-clique can be compressed to its actual c

Excluding sixteen-vertex critical cores in the order-twenty Earth–Moon problem

We prove that a $K_9$-free graph on twenty vertices with at most 108 edges and minimum degree at least nine is nine-colorable whenever an independent triple is

Private eight-cliques of degree nine are reducible in biplanar graphs

We prove that a biplanar graph containing an eight-clique whose vertices have at most two outside neighbors admits a smaller biplanar replacement on the remaini

Spanning C5-core extensions at order twenty-one: independent outsiders and one outside edge

We study order-twenty-one ten-chromatic obstructions to the Earth–Moon problem through seventeen-vertex spanning clique blow-ups of $C_5$. No biplanar ten-criti

Planar smoothing and private clique reductions in biplanar nine-coloring

We prove that every private seven-clique whose vertices have total degree at most nine is reducible in a minimum-order counterexample to nine-colorability of bi

Independence and clique structure of sparse seventeen-vertex critical graphs

We prove that every $K_9$-free 9-critical graph on seventeen vertices with at most $81$ edges has independence number at most three, and exhibit an $82$-edge cr

Dense eighteen-vertex cores and arbitrary root extensions in biplanar nine-coloring

We study nine-coloring extensions over an eighteen-vertex biplanar graph $H$ with independence number at most two. We prove that if $e(H)\leq90$, then adding ex

Dense eight-clique attachments in biplanar nine-coloring

We study biplanar graphs through their attachments to a specified eight-clique. We prove that every such graph is nine-colorable if each outside vertex has at l

Degree-minus-one list coloring of two-fan cores with even paths

We study local list-coloring reductions for the Earth–Moon problem. Let $J(a,b)$ consist of adjacent hubs $x,y$, two internally disjoint $A$–$B$ paths of length

Induced-interface amalgams of a nine-critical biplanar graph

We study constructions that identify induced subgraphs of copies of a known seventeen-vertex, 82-edge nine-critical biplanar graph. Any number of copies sharing

Rigidity of an eighteen-vertex Earth–Moon counterexample

We study the eighteen-vertex boundary of the Earth–Moon problem. We prove that a biplanar 10-critical graph on eighteen vertices, if one exists, has exactly 96

Matching palettes and four-terminal reductions for a skeletal biplanar template

We study biplanar graphs partitioned into three core cliques and a remainder of independence number at most two, where remainder vertices miss exactly one core

Selected-color completion and sharp boundary extension in biplanar graphs

We study extension of prescribed nine-colorings in biplanar graphs using planar completion of selected color classes. For two layers inside disjoint quadrilater

Common interfaces and literal planar stars of a nine-color kernel

We determine two assembly closures for an explicit nineteen-vertex, ninety-four-edge nine-chromatic biplanar kernel consisting of a six-clique joined to a five-

Uniform list coloring of long odd two-fans and rigidity in critical graphs

We study list-coloring extensions and exterior rigidity for the Earth–Moon problem. The graph $J(a,b)$ consists of adjacent hubs and two internally disjoint odd

Degree-minus-one coloring of contacted J(5,5) patches

We prove a finite local extension theorem for the contacted odd/odd two-fan patches appearing in a 492-vertex biplanar occurrence barrier. Every J(5,5) with one

Coordinated center boundaries in ordinary covers of G25

We study ordinary covers of the ten-chromatic graph $G_{25}$ as possible sources of ten-chromatic biplanar graphs. Four earlier extension results are retained.

Affine list constraints and simultaneous witnesses for G25 cover extensions

We study signed list constraints and prescribed-center extensions in ordinary covers of the ten-chromatic graph $G_{25}$. Our affine theorem partitions variable

Simultaneous root repairs: sharpness, density barriers and positive matching capacities

We study simultaneous coloring repairs for a root clique and an isolated root over specified cores. An explicit eleven-vertex core proves the abstract four-root

A biplanar graph with no singleton neighborhood-identification reduction for nine colors

A natural route to nine-coloring a biplanar graph deletes one vertex and identifies independent blocks of its neighborhood until at most eight boundary vertices

Sparse exterior extensions of dense nine-critical cores

Let $G$ be biplanar and let $Q$ be an induced nine-vertex-critical subgraph of order at most seventeen with $e(Q)\ge6|V(Q)|-20$. We prove that $G$ is nine-color

Created triads and matching palettes in biplanar antiprismatic thickenings

Different changeable edges in an antiprismatic thickening can create disjoint families of independent triples, so a reduction based on deleting one selected tri