Cycle Partition of Two-Connected and Two-Edge Connected Graphs

Abstract

Let G be any graph. Then c(G) denotes the circumference of G and is defined as follows: if G is edgeless then c(G) = 1; G is acyclic but G contains at least one edge, then c(G) = 2, if G contains a cycle then c(G) denotes the length of a longest cycle in G. A graph G is called (c ₁, c ₂)-partitionable for a pair of positive integers (c ₁, c ₂), if c ₁ + c ₂ = c(G) and the vertex set V(G) admits a partition (V ₁, V ₂) such that ${c(\langle V_i \rangle) \leq c_i}$, i = 1, 2. A graph G is called c-partitionable if G is (c ₁, c ₂)-partitionable for every pair of positive integers (c ₁,c ₂) satisfying c ₁ + c ₂ = c(G). Recently Nielsen (Discret Math 308:6339–6347, 2008) conjectured that every graph is c-partitionable. Nielsen’s conjecture is a cycle version of the well-known long standing Path Partition Conjecture introduced by Lovasz and Mihok in 1981 and studied in the thesis of Hajnal (Graph partitions. Thesis, J.A. University, Szeged, 1984). In this paper we show that Nielsen’s conjecture is true for a class of two-connected and two-edge connected graphs having a special ear decomposition. Inspired by the ${\tau}$ -partition and c-partition, we introduced α-partitionable graphs and ω-partitionable graphs. A graph G is called α-partitionable, if there exist a partition (V ₁, V ₂) of V(G) such that ${\alpha(\langle V_1 \rangle) \leq \alpha_1}$ and ${\alpha(\langle V_2 \rangle) \leq \alpha_2}$ for every pair of positive integers (α ₁, α ₂) with ${\alpha_1 + \alpha_2 = \alpha(G)}$, where α(G) is the cardinality of maximum independent set of a graph G. Here, we show that every graph is α-partitionable. Similarly, ω-partitionable graph is defined, where ω(G) is the number of vertices of maximum complete subgraph of a graph G. We proved that every perfect graph is ω-partitionable and we discuss a related open problem.