Nowhere-zero 3-flows and modulo k-orientations

Exponentially many nowhere-zero Z3-, Z4-, and Dvořák1, Mohar2, Šámal3 2017Electronic Notes in Discrete Mathematics1000View full textAdd to dashboardCite

“…Every 4-edge-connected graph has a nowhere-zero Z 2 × Z 2 -flow. Theorem 1.6 (Lovász, Thomassen, Wu, and Zhang [5]). Every 6-edge-connected graph has a nowhere-zero Z 3 -flow.…”

“…Unlike the previous sections, where we prove everything from first principles, here we rely on the result of L. M. Lovász et al [5]. Another essential part is a new splitting lemma that allows us to handle vertices of degree 6.…”

“…We will use a result of L. M. Lovász et al, who proved in [5] that every 6-edge-connected graph admits a nowhere-zero Z 3 -flow. To state their result precisely, we need to introduce a generalization of flows on graphs and a delicate way to measure graph connectivity.…”

“…For a set X ⊆ V (G), let β(X) = x∈X β(x) and let deg(X) denote the number of edges of G with exactly one end in X. Let τ (X) be defined as follows: Note that we use τ where the authors of [5] have 4 + τ . Also, the formulation in [5] is slightly stronger, which is essential for the inductive proof of their main result, but irrelevant for us.…”

Partitioning Perfect Graphs into Stars

“…While in this work we study the H-Partition problem, which partitions the vertex set of a graph into mutually vertex-disjoint copies of some fixed pattern graph H, the literature also studies the H-Decomposition problem, which partitions the edge set of a graph into mutually edge-disjoint copies of a pattern H. In general, H-Decomposition is NP-hard [8], yet easy to solve on highly-connected graphs if H is a k-star: Thomassen [32] shows that every (k 2 + k)-edge-connected graph has a k-star decomposition provided its number of edges is a multiple of k. Lovász et al [21] strengthen this result to (3k − 3)-edge-connected graphs for odd k ≥ 3. However, since a graph may have a k-star decomposition without having a k-star partition and vice versa, the results on H-Decomposition are not applicable to the Star Partition problem considered in our work.…”