Minor stars in plane graphs with minimum degree five

The weight of a subgraph $H$ in $G$ is the sum of the degrees in $G$ of vertices of $H$. The {\em height} of a subgraph $H$ in $G$ is the maximum degree of vertices of $H$ in $G$. A star in a given graph is minor if its center has degree at most five in the given graph. Lebesgue (1940) gave an approximate description of minor $5$-stars in the class of normal plane maps with minimum degree five. In this paper, we give two descriptions of minor $5$-stars in plane graphs with minimum degree five. By these descriptions, we can extend several results and give some new results on the weight and height for some special plane graphs with minimum degree five.

💡 Research Summary

The paper investigates “minor 5‑stars” in plane graphs whose minimum vertex degree is five. A minor k‑star is a star whose centre has degree at most five; the weight of a subgraph is the sum of the degrees (in the original graph) of its vertices, and its height is the maximum degree among those vertices. Earlier work (Lebesgue 1940, Franklin 1922, Borodin‑Ivano​va, Jendrol’‑Madaras, etc.) gave partial descriptions of such stars and bounds on their weight, typically of the form ΩΔ ≤ Δ + 30 for a graph with maximum degree Δ. The authors aim to provide a more detailed, cyclic‑order‑aware description of the neighbours of a 5‑vertex and to use this to obtain sharper and more general results.

Two families of ordered stars are introduced. The first, denoted h_{κ1,κ2,κ3,κ4,κ5}, is a star whose centre has degree five and whose five neighbours, taken in clockwise order around the centre, have degrees at most κ1,…,κ5 respectively. The second family is a variant that relaxes some ordering constraints. By enumerating concrete tuples (κ1,…,κ5) the authors obtain exhaustive lists of possible minor 5‑stars that must appear in any plane graph with minimum degree five.

The main result (Theorem 1) states that every such plane graph contains at least one of 52 explicitly listed ordered 5‑stars, for example h_{5,7,7,5,17}, h_{8,5,5,11,6}, h_{6,6,6,6,11}, etc. From this single theorem the authors immediately derive several known results (Theorems 2–7), such as the existence of a minor 5‑star of weight at most Δ + 29 when Δ ≥ 13, the existence of a minor 4‑star of weight at most 30, and the classic (5,6)‑edge, (6,5,6)‑path, and (6,6,6)‑star statements. Additional theorems (8–11) treat special cases where certain degree ranges are forbidden, showing that even under these restrictions a limited set of ordered stars must appear.

The proof relies on the classic discharging method. Each vertex v receives an initial charge μ(v)=deg(v)−6, and each face f receives μ(f)=2·deg(f)−6. Since the total initial charge equals –12 (by Euler’s formula), the goal is to redistribute charge according to a carefully crafted set of rules (R1–R7) so that every vertex ends with non‑negative final charge μ′(v). The rules distinguish several neighbour types: strong 5‑neighbors (a 5‑vertex whose two adjacent neighbours are 6⁺), non‑strong 5‑neighbors, weak 5‑neighbors, and twice‑weak 5‑neighbors. High‑degree vertices (7,8,10,11,13,14) send fractional amounts of charge to adjacent 5‑vertices; 5‑vertices may also return charge to a high‑degree neighbour under specific configurations. Detailed case analysis shows that if none of the listed ordered stars were present, some vertex would end with negative charge, contradicting the total sum. Hence at least one of the listed stars must exist.

A second exhaustive list (Theorem 12) provides 58 ordered stars of a slightly different form, again guaranteeing their presence in any minimum‑degree‑5 plane graph. Using this list the authors prove further consequences: Theorem 13 improves the weight bound to Δ + 28 for Δ ≥ 16; Theorem 14–16 give weight and height bounds (45, 44, 51 respectively) under the exclusion of certain ordered stars, thereby sharpening earlier results of Borodin and Ivano​va.

The paper emphasizes that the results hold for all plane graphs with minimum degree five, not only for 3‑connected planar graphs (3‑polytopes), thus covering a broader class than the previously studied P₅. The authors also note that their ordered‑star descriptions provide a more precise tool for the theory of light subgraphs in planar graphs, and they suggest that similar techniques could be applied to other degree‑restricted families.

In summary, the work delivers a comprehensive, ordered‑star framework for minor 5‑stars in planar graphs with minimum degree five, refines existing weight and height bounds, and extends the applicability of light‑subgraph methods beyond the traditionally studied 3‑connected case. This contributes a valuable structural insight to planar graph theory and offers a solid foundation for future investigations of degree‑constrained substructures.