Group Connectivity and Modulo Orientations of Graphs

Abstract

This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte's 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte's 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowherezero 3-flow. Jaeger et al. (1992) further conjectured that every 5-edge-connected graph is Z 3-connected, whose truth implies the 3-Flow Conjecture. Extending Tutte's flows conjectures, Jaeger's Circular Flow Conjecture (1981) says every 4p-edge-connected graph admits a modulo (2p + 1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo 2p + 1 at every vertex. Note that the p = 1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of p = 2 implies the 5-Flow Conjecture. This work is devoted to provide some partial results on these problems. It is proved in Chapter 2 that every graph with four edge-disjoint spanning trees is Z 3connected. Consequently, Jaeger et al.'s group connectivity conjecture and Tutte's 3-Flow Conjecture hold for 5-edge-connected essentially 23-edge-connected graphs. We also provide several equivalent versions of Jaeger et al.'s group connectivity conjecture and indicate that it is enough to verify the conjecture for 5-edge-connected essentially 8-edge-connected graphs. In Chapter 3, Tutte's 3-Flow Conjecture is verified for graphs with independence number at most 4. The relation of orientation and group connectivity is studied in Chapter 4. It shows that every strongly Z m-connected graph contains m − 1 edge-disjoint spanning trees, and hence every Z m-connected graph G has (m − 1)(|V (G)| − 1)/(m − 2) edges, which solves a conjecture of Luo et al. (2012). Those results are applied to establish some monotonicity properties of group connectivity that every strongly Z 5-connected graph is Z 3-connected, and every Z 3-connected graph is A-connected for any Abelian group A with size |A| ≥ 4. Infinite families of counterexamples to Jaeger's Circular Flow Conjecture are presented in Chapter 5. For p ≥ 3, there are 4p-edge-connected graphs not admitting modulo (2p + 1)orientation; for p ≥ 5, there are (4p + 1)-edge-connected graphs not admitting modulo (2p + 1)orientation. Towards the p = 2 case of Circular Flow Conjecture and the 5-Flow Conjecture, we show in Chapter 6 that every 10-edge-connected planar graph admits a modulo 5-orientation.

Weighted Modulo Orientations of Graphs and Signed Graphs

“…(Lovász et al [20]) Every 6k-edge-connected graph G admits a b-orientation for any Z 2k+1 -boundary b of G. Theorem 3. (Han et al [7] and Li [16]) (i) If k 3, then there exist 4k-edge-connected graphs admitting no mod (2k + 1)orientation. (i) If k 5, then there exist (4k + 1)-edge-connected graphs admitting no mod (2k + 1)orientation.…”

Flow Extensions and Group Connectivity with Applications

“…Now we prove Theorem 2 using similar constructions, but employing Lemma 1(ii) instead. The argument presented here is a slight modification of that in the author's Ph.D dissertation [8].…”

“…Proof of Theorem 2: The relations of some of those statements have been investigated in [3]. The proofs of "(b-i)⇔(b-ii)" and "(b-ii)⇒(c)⇒(a)" have been presented in [3,8]. Clearly, we also have "(d)⇒(a)".…”