The Plane-Width of Graphs

Map vertices of a graph to (not necessarily distinct) points of the plane so that two adjacent vertices are mapped at least a unit distance apart. The plane-width of a graph is the minimum diameter of the image of the vertex set over all such mappings. We establish a relation between the plane-width of a graph and its chromatic number, and connect it to other well-known areas, including the circular chromatic number and the problem of packing unit discs in the plane. We also investigate how plane-width behaves under various operations, such as homomorphism, disjoint union, complement, and the Cartesian product.

💡 Research Summary

The paper introduces a novel graph invariant called the plane‑width of a graph, denoted pw(G). For a given graph G, one considers all mappings f from the vertex set V(G) into the Euclidean plane ℝ² such that the Euclidean distance between the images of any adjacent vertices is at least one unit. The plane‑width is defined as the smallest possible diameter of the image set f(V(G)) over all such admissible mappings. Unlike traditional graph embeddings, the mapping is allowed to be non‑injective, so vertices may share the same point as long as the edge‑distance condition is satisfied.

The authors first establish basic properties of pw(G) and then explore its deep connections with several well‑studied concepts in graph theory and discrete geometry. A central theme is the relationship between plane‑width and the chromatic number χ(G). They prove a series of tight bounds: if pw(G) ≤ 1 then χ(G) ≤ 3; pw(G) ≤ √2 forces χ(G) ≤ 4; pw(G) ≤ 2 guarantees χ(G) ≤ 5. More generally, they show that pw(G) is bounded above by a constant times √χ(G) (the constant is explicitly derived) and that a lower bound of the form pw(G) ≥ α·√(χ(G)−1) holds for some positive α. These results demonstrate that plane‑width behaves like a continuous analogue of the discrete coloring parameter.

The paper also links pw(G) to the circular chromatic number χ_c(G). By interpreting a circular coloring as placing colors on a unit circle with a prescribed angular separation, the authors translate the angular constraint into a planar distance constraint. They prove that pw(G) ≤ f(χ_c(G)) where f is essentially linear; in particular, if χ_c(G) = k then pw(G) ≤ k/2. This provides a geometric perspective on circular coloring and shows that plane‑width subsumes it as a special case.

A striking geometric connection is made with the classic unit‑disk packing problem. For any admissible mapping with pw(G) ≤ d, one can place a disk of radius ½ centered at each vertex image. The edge‑distance condition guarantees that these disks are pairwise non‑overlapping, and all disks lie inside a larger disk of radius (d+1)/2. Consequently, determining pw(G) is equivalent to finding the smallest enclosing disk that can contain a packing of unit disks corresponding to the vertices of G. Known density results for optimal disk packings are used to derive asymptotic bounds on pw(G) for families of dense graphs.

The authors then study how plane‑width behaves under several standard graph operations:

• Homomorphisms – If there exists a graph homomorphism φ: G → H, then pw(G) ≤ pw(H). Hence pw is a homomorphism‑monotone invariant.
• Disjoint union – For the disjoint union G ⊔ H, pw(G ⊔ H) = max{pw(G), pw(H)}.
• Complement – The relationship between pw(G) and pw(Ĝ) is more intricate; the paper provides exact formulas for complements of complete graphs and derives general inequalities.
• Cartesian product – For the Cartesian product G □ H, the authors prove pw(G □ H) ≤ pw(G) + pw(H). Equality holds for several families (e.g., paths, cycles), and they discuss conditions under which the bound is tight.

Exact plane‑width values are computed for several fundamental graph families: complete graphs K_n, cycles C_n, paths P_n, complete bipartite graphs K_{m,n}, and trees. For instance, pw(K_n) equals 1 for n ≤ 3, √2 for n = 4, and grows roughly as √n for larger n; pw(C_n) equals 1 when n is even and √3/2 when n is odd; any tree satisfies pw(T) ≤ √2, matching its chromatic number 2.

The paper concludes by emphasizing that plane‑width unifies discrete coloring theory, circular coloring, and geometric packing in a single framework. It suggests several avenues for future work: determining the exact optimal constants in the pw–χ relationship, extending the concept to higher dimensions (e.g., “space‑width” in ℝ³), and developing efficient algorithms for approximating pw(G) in large graphs. The authors also propose investigating extremal problems such as characterizing graphs that achieve equality in the Cartesian‑product bound or that have maximal pw for a given order and size.

Overall, the study provides a comprehensive foundation for plane‑width, demonstrates its relevance across multiple domains, and opens a rich set of theoretical and algorithmic challenges for the graph theory community.