Coloring graphs using topology (1412.6985v1)

Abstract: Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces we can adapt the degrees along the boundary of that surface. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal colouring would imply the four colour theorem. The method is expected to give a reason "why 4 colours suffice" and suggests that every two dimensional geometric graph of arbitrary degree and orientation can be coloured by 5 colours: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to colour it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3,4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. We have implemented and illustrated all the topological aspects described in this paper on a computer. So far we still need human guidance or simulated annealing to do the refinements in the higher dimensional host graph.

• The paper's main contribution is the introduction of a topological refinement method using graph cobordism to achieve four-coloring of two-dimensional graph boundaries.
• It presents a methodology where ensuring even interior edge degrees in a three-dimensional graph aids in applying four-coloring techniques to its planar boundary.
• The work conjectures that every two-dimensional geometric graph can be colored with five colors, paving the way for future research in automated graph coloring algorithms.

Coloring Graphs Using Topology

The paper "Coloring Graphs Using Topology" by Oliver Knill provides a novel approach to understanding the coloring of graphs with implications towards the four-color theorem in graph theory. The work synthesizes concepts from both graph theory and topology to extend the discussion of graph coloring beyond two-dimensional planar graphs to higher-dimensional contexts.

At its core, the paper investigates how higher-dimensional graphs facilitate the coloring of two-dimensional geometric graphs. Specifically, if a two-dimensional graph $G$ is the boundary of a three-dimensional graph $H$, Knill suggests a refinement technique within $H$ that allows for coloring $G$ with four colors. This method hinges on refining $H$ such that all interior edge degrees are even, thereby facilitating a four-coloring of the boundary $G$.

The main thrust is a topological approach where the notion of graph cobordism is key. The concept is extended to a graph being self-cobordant via another higher-dimensional host graph that serves as a discretization product of $G$ with an interval. This involves the notion that, due to the Euler curvature being zero in such three-dimensional geometric graphs, the odd degree edge set forms cycles which—if $H$ is simply connected—are also boundaries.

An essential result from this paper is the conjecture that every two-dimensional geometric graph could be colored by five colors. This conjecture is motivated by considerations of the projective plane being non-boundary in a three-dimensional graph context and the lack of simple connectivity for higher genus surfaces. The results provide not only theoretical insights but also pave a conjectural upper bound for the chromatic number of such graphs to be five in complex scenarios.

In alignment with topological concepts, Knill demonstrates using Fisk's method how for each surface type—characterized by genus and orientation—there exist examples with chromatic numbers three, four, or five. These illustrative examples support the theoretical framework laid out in the paper.

Moreover, while the implementation of these ideas on computers is acknowledged in the paper, Knill clearly states the current limitations, highlighting the reliance on manual guidance for complex topological manipulations within high-dimensional host graphs. This imposes an opportunity for future work to develop automated algorithms or refine existing methods of graph coloring through computational means.

The implications of this work reach both practical and theoretical domains, proposing a potentially new method for proving the four-color theorem or its higher-dimensional analogs through graph cobordism. Speculatively, if these methods are successful, they could lead to a more algorithmic understanding of complex graph colorings and have implications in fields requiring network embeddings in higher-dimensional spaces.

Finally, the paper prompts several open questions and conjectures concerning higher-dimensional graph embeddings, offering a fresh perspective and a directional guide for researchers looking to explore the intricate intersection of graph theory and topology.