Nowhere-Zero Unoriented 6-Flows on Certain Triangular Graphs

A nowhere-zero unoriented flow of graph G is an assignment of non-zero real numbers to the edges of G such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A nowhere-zero unoriented k-flow is a flow with values from the set {±1, . . ., ±(k − 1)}, for short we call it NZ-unoriented k-flow. Let H₁ and H₂ be two graphs, H₁⊕H₂ denote the 2-sum of H₁ and H₂, if E(H₁⊕H₂) = E(H₁) ∪ E(H₂), |V(H₁)∩V(H₂)|=2, and |E(H₁)∩E(H₂)| = 1. A triangle-path in a graph G is a sequence of distinct triangles T₁, T₂, . . ., T_m in G such that for 1 ≤ i ≤ m, |E(T_i)∩E(T_i+1)| = 1 and E(T_i)∩E(T_j)=∅ if j>i+1. A triangle-star is a graph with triangles such that each triangle having one common edges with other triangles. Let G be a graph which can be partitioned into some triangle-paths or wheels H₁, H₂, . . ., H_t such that G = H₁⊕H₂⊕...⊕H_t. In this paper, we prove that G except a triangle-star admits an NZ-unoriented 6-flow. Moreover, if each H_i is a triangle-path, then G except a triangle-star admits an NZ-unoriented 5-flow.