Nowhere-zero 3-flows in triangularly connected graphs

Abstract

Let H 1 and H 2 be two subgraphs of a graph G. We say that G is the 2-sum of H 1 and H 2 , denoted byA connected graph G is triangularly connected if for any two edges e and e , which are not parallel, there is a triangle-path T 1 T 2 · · · T m such that e ∈ E(T 1 ) and e ∈ E(T m ). Let G be a triangularly connected graph with at least three vertices. We prove that G has no nowhere-zero 3-flow if and only if there is an odd wheel W and a subgraph G 1 such that G = W ⊕ 2 G 1 , where G 1 is a triangularly connected graph without nowhere-zero 3-flow. Repeatedly applying the result, we have a complete characterization of triangularly connected graphs which have no nowhere-zero 3-flow. As a consequence, G has a nowhere-zero 3-flow if it contains at most three 3-cuts. This verifies Tutte's 3-flow conjecture and an equivalent version by Kochol for triangularly connected graphs. By the characterization, we obtain extensions to earlier results on locally connected graphs, chordal graphs and squares of graphs. As a corollary, we obtain a result of Barát and Thomassen that every triangulation of a surface admits all generalized Tutte-orientations.

A low-polynomial algorithm for assembling clusters of orthologous groups from intergenomic symmetric best matches

“…The main idea in our approach is to find a specific class of subgraphs, recently called triangularly connected subgraphs (Fan et al , 2008), by iterating over edges instead of triangles, as explained in more detail below.…”

Nowhere-zero 3-flows and Z3-connectivity of a family of graphs

“…. , C k such that e 1 ∈ E(C 1 ) and and H 2 be two subgraphs of a graph G. We say that G is the 2-sum of H 1 and H 2 , denoted by [2][3][4]7,8]). Let A be an abelian group with |A| ≥ 3.…”

Realizing degree sequences as Z3-connected graphs

“…Contracting this wheel W 4 and contracting the 2-cycle generated in the process, we get K 1 . By parts (1), (3), (5) and (6) Lemma 2.6 shows that there exists a nowhere-zero…”

“…(4) K m,n is A-connected if m ≥ n ≥ 4; neither K 2,t (t ≥ 2) nor K 3,s (s ≥ 3) is Z 3 -connected; (5) Each even wheel is Z 3 -connected and each odd wheel is not; (6) Let H ⊆ G and H be A-connected. G is A-connected if and only if G/H is A-connected; (7) If G is not A-connected, then any spanning subgraph of G is not A-connected.…”