[Paper Review] A Proof Technique for Skewness of Graphs

Gek L. Chia, Chan L. Lee+1 more|arXiv (Cornell University)|Sep 19, 2017

Advanced Graph Theory Research6 references3 citations

TL;DR

This paper introduces a novel weighted edge technique to prove the skewness of generalized Petersen graphs P(4k,k) for odd k ≥ 9, establishing that μ(P(4k,k)) = k+2. By assigning strategic weights to edges and analyzing face weights in planar subgraphs, the authors resolve a long-standing conjecture from [3] using a new proof framework based on Euler’s formula and edge-weight inequalities.

ABSTRACT

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. By appropriately introducing a weight to each edge of a graph, we determine, among other thing, the skewness of the generalized Petersen graph P(4k, k) for odd k at least 9. This provides an answer to the conjecture raised in [3].

Motivation & Objective

• To resolve the conjecture that μ(P(4k,k)) = k+2 for odd k ≥ 5, extending prior results for even k.
• To develop and apply a novel weighted edge method for skewness analysis in non-planar graphs.
• To establish a general formula for the skewness of the Qs(k) family of graphs, μ(Qs(k)) = ⌈(s−2)k/2⌉ + 1 for k ≥ 4.
• To demonstrate that the skewness of P(4k,k) cannot be k+1 by contradiction using face weight and vertex independence constraints.
• To provide a unified proof technique based on edge weighting and Euler’s formula, applicable to generalized Petersen and related graphs.

Proposed method

• Assigns edge weights: 4 to outer cycle edges u_iu_{i+1}, k−3 to spokes u_iv_i, and 2k−2 to the inner cycle edges v_iv_{i+k} in P(4k,k).
• Defines total weight W(G) = 4k(3k−1) for P(4k,k), and uses weight conservation in planar subgraphs H obtained by removing t edges.
• Applies Euler’s formula to derive a lower bound on t using the inequality 2(W(H)) ≥ (8k−8)(4k−t+2), leading to t ≥ k+1.
• Introduces a refined weight function w'(e) with values 4, 1, and 2 on edges of types u_iu_{i+1}, u_iv_i, and v_iv_{i+k} respectively, to analyze face types in H.
• Uses face type classification (i)–(iv) to show that only specific cycles can appear in planar subgraphs when equality is tight.
• Reduces the problem to a subgraph isomorphic to Q₃(k), leveraging known skewness μ(Q₃(k)) = (k+3)/2 to derive a contradiction if μ(P(4k,k)) = k+1.

Experimental results

Research questions

• RQ1 What is the skewness of the generalized Petersen graph P(4k,k) for odd k ≥ 9?
• RQ2 Can a new weighted edge technique be developed to prove skewness bounds in non-planar graphs?
• RQ3 Is the conjecture μ(P(4k,k)) = k+2 true for odd k ≥ 5, and can it be proven without relying on prior methods?
• RQ4 What structural constraints arise in a planar subgraph H of P(4k,k) when the skewness is assumed to be k+1?
• RQ5 How does the skewness of Qs(k) relate to that of P(4k,k), and can Qs(k) serve as a building block for proving skewness in generalized Petersen graphs?

Key findings

• The skewness of P(4k,k) is exactly k+2 for all odd k ≥ 9, confirming the conjecture from [3].
• The skewness of the Qs(k) family is μ(Qs(k)) = ⌈(s−2)k/2⌉ + 1 for k ≥ 4, generalizing earlier results for s = 3 and s = 4.
• A planar subgraph H of P(4k,k) with μ(P(4k,k)) = k+1 would require only specific face types (i)–(iv), and the edge set R(P(4k,k)) must consist only of independent outer cycle edges.
• The existence of two disjoint faces of type (i) in H implies at least 2k−2 good vertices, leading to at least k+1 consecutive bad vertices in the outer cycle.
• The subgraph induced by removing k−1 vertices and edges from P(4k,k) is a subdivision of Q₃(k), and since μ(Q₃(k)) = (k+3)/2, the total skewness exceeds k+1.
• The contradiction proves that μ(P(4k,k)) ≥ k+2, and since the upper bound was already known, equality holds: μ(P(4k,k)) = k+2.