Edge-disjoint odd cycles in 4-edge-connected graphs

Abstract

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of the difficulties of this problem is that the Erdős-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f (k) satisfying the following: For any 4-edge-connected graph G = (V, E), either G has edge-disjoint k odd cycles or there exists an edge set F ⊆ E with |F | ≤ f (k) such that G − F is bipartite. We note that the 4-edge-connectivity is best possible in this statement. Similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edgeconnected graph with n vertices. We show that, for any ε > 0, if k = O((log log log n) 1/2−ε ), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n. ACM Subject Classification G.2.2 Graph Theory

An edge variant of the Erdős–Pósa property

“…In this direction, it is known that f M(J) = O(k) in the case where graphs are restricted to some non-trivial minor-closed class [10].We consider the edge counterpart of the Erdős-Pósa property, where packings are edgedisjoint (instead of vertex-disjoint) and coverings contain edges instead of vertices. We say that a graph class G satisfies the edge variant of the Erdős-Pósa property if there exists a function f G such that, for every graph G and every positive integer k, either G contains k mutually edge-disjoint subgraphs, each isomorphic to a graph in G, or it contains a set X of f G (k) edges meeting every subgraph of G that is isomorphic to a graph in G. Recently, the edge variant of the Erdős-Pósa property was proved in [12] for 4-edge-connected graphs in the case where G contains all odd cycles. J (G) for the minimum size of a subset S ⊆ E(G) (called J-edge-hitting set) that meets the edge sets of all models of J in G. Obviously, for every two graphs G and J, the following inequality holds:A graph J is said to satisfy the (vertex-)Erdős-Pósa property for minors (vertex-Erdős-Pósa property for short) if there is a function f J : N → N, called vertex-Erdős-Pósa gap of J, such that for every graph G, the following holds:The research of this paper is motivated by the course of detecting graphs J for which…”

“…We consider the edge counterpart of the Erdős-Pósa property, where packings are edgedisjoint (instead of vertex-disjoint) and coverings contain edges instead of vertices. We say that a graph class G satisfies the edge variant of the Erdős-Pósa property if there exists a function f G such that, for every graph G and every positive integer k, either G contains k mutually edge-disjoint subgraphs, each isomorphic to a graph in G, or it contains a set X of f G (k) edges meeting every subgraph of G that is isomorphic to a graph in G. Recently, the edge variant of the Erdős-Pósa property was proved in [12] for 4-edge-connected graphs in the case where G contains all odd cycles. J (G) for the minimum size of a subset S ⊆ E(G) (called J-edge-hitting set) that meets the edge sets of all models of J in G. Obviously, for every two graphs G and J, the following inequality holds:…”

An $$O(\log \mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$\theta _r$$ θ r

“…Interestingly, not much more is known on the graphs H for which C H has the e-EP-property and is tempting to conjecture that the planarity of H provides again the right dichotomy. Other graph classes that are known to have the e-EP-property are rooted cycles [29] (here the cycles to be covered and packed are required to intersect some particular set of terminals of G) and odd cycles for the case where G is a 4-edge connected graph [23], a planar graph [26], or a graph embeddable in an orientable surface [24].…”

Packing and Covering Immersions in 4-Edge-Connected Graphs

“…On the other hand, even though a family of graphs does not have the edge-variant of the Erdős-Pósa property, this family possibly has this property if we restrict the host graphs to be members of a smaller class of graphs. For example, the set of odd cycles does not have the edge-variant of the Erdős-Pósa property, but Kawarabayashi and Kobayashi [9] proved that it has the edge-variant of the Erdős-Pósa property in 4-edge-connected graphs. We address the same direction in this paper and prove that for every graph H, I(H) has the edge-variant of the Erdős-Pósa property in 4-edge-connected graphs.…”