Six Signed Petersen Graphs

Abstract

The Petersen graph with signed edges makes a fascinating example of many aspects of signed graph theory. I will show that there are exactly six essentially different ways to sign P, find their automorphisms, show some ways in which their signs matter, color them, and mention two applications of signed graphs in which signed Petersen graphs have not yet made an appearance. The Petersen graph is P = (V, E) with vertex set V = {vij: 1 ≤ i < j ≤ 5} and edge set E = {vijvkl: {i, j} ∩ {k, l} = ∅}. A signed graph is a pair Σ: = (Γ, σ) where Γ is a graph and σ: E(Γ) → {+1, −1} is an (edge) signature that labels each edge positive or negative. Hence, a signed Petersen graph is (P, σ); two examples are +P: = (P, +1), where every edge is positive, and −P: = (P, −1), where every edge is negative. The sign of a circle (cycle, circuit, polygon) C is σ(C): = the product of the signs of the edges in C. The most essential fact about a signed graph is the set of circles that have negative sign. If this set is empty we call the signed graph balanced. Such a signed graph is equivalent to an unsigned graph in most ways. Harary [4] introduce

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