A Note on Arboricity of 2-edge-connected Cubic Graphs
نویسندگان
چکیده
The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well known that a(G) ≤ 3 for any planar graph G, and that a(G) ≤ 2 for any planar graph G of diameter at most 2. The conjecture that every planar graph G without 3-cycles has a vertex partition (V1, V2) such that V1 is an independent set and V2 induces a forest was given in [European J. Combin., 2008, 29(4): 1064-1075]. In this paper, we prove that a 2-edge-connected cubic graph which satisfies some condition has this partition. As a corollary, we get the result that every up-embeddable 2-edge-connected cubic graph G (G = K4) has a vertex partition (V1, V2) such that V1 is an independent set and V2 induces a forest.
منابع مشابه
A note on Fouquet-Vanherpe’s question and Fulkerson conjecture
The excessive index of a bridgeless cubic graph $G$ is the least integer $k$, such that $G$ can be covered by $k$ perfect matchings. An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless cubic graph has excessive index at most five. Clearly, Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5, so Fouquet and Vanherpe as...
متن کاملOn linear arboricity of cubic graphs
A linear forest is a graph in which each connected component is a chordless path. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. When each path has length at most k a linear forest is a linear k-forest and lak(G) will denote the minimum number of linear k-forests partitioning E(G). We cl...
متن کاملA note on vertex-edge Wiener indices of graphs
The vertex-edge Wiener index of a simple connected graph G is defined as the sum of distances between vertices and edges of G. Two possible distances D_1(u,e|G) and D_2(u,e|G) between a vertex u and an edge e of G were considered in the literature and according to them, the corresponding vertex-edge Wiener indices W_{ve_1}(G) and W_{ve_2}(G) were introduced. In this paper, we present exact form...
متن کاملOn isomorphic linear partitions in cubic graphs
A linear forest is a graph that connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G) = 2 when G is a cubic graph and Wormald [17] conjectured that if |V (G)| ≡ 0 (mod 4), then it is always possible to find a linear partition...
متن کاملON THE EDGE COVER POLYNOMIAL OF CERTAIN GRAPHS
Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...
متن کامل