On Simultaneous 2-locally-balanced 2-partition for Two Forests with Same Vertices

The existence of a partition of the common set of the vertices of two forests into two subsets, when difference of their capacities in the neighbourhood of each vertex of each forest not greater than 2 is proved, and an example, which shows that improvement of the specified constant is impossible is brought. In this paper we continue researches started in [1-2], devoted to locally-balanced partitions of a graph. We consider undirected graphs and multigraphs without loops. The set of vertices of the multigraph G is denoted by V (G), and the set of edges of G-by E(G), and the maximum degree of the vertices of G-by ∆(G). The eccentricity of a vertex v ∈ V (G) is denoted by exG(v). Non-defined concepts can be found in [3]. For v ∈ V (G) we shall define sets γG(v) = {w ∈ V (G)/(w, v) ∈ E(G)} and ηG(v) = {e ∈ E(G)/v incident to e}. A function f : M → {0, 1} is called 2-partition of a finite set M . If f is a 2-partition of a finite set M , then for ∀M0 ⊆M we define the number bf (M0) as follows: bf (M0) = ||{m ∈M0/f(m) = 1}| − |{m ∈M0/f(m) = 0}||. Let G1and G2 are undirected graphs without loops with V (G1) = V (G2) ≡ V . The 2partition f of the set V is called simultaneous k-locally-balanced (k ∈ Z, k ≥ 0) 2-partition of the graphs G1and G2 if: max i=1,2 max v∈V bf (γGi(v)) = k. Let D is a tree, and let v1(D) ∈ V (D) is an arbitrarily chosen vertex. For i = 0, 1, ..., exD(v1(D)) we define a subset Si ⊆ V (D) as follows: Si ≡ {w ∈ V (G)/ρ(w, v1(G)) = i}. For i = 1, 2, ..., exD(v1(D)) and u ∈ Si−1 let’s define Si(u) ≡ {w ∈ Si/(w, u) ∈ E(D)}. We define a family of subsets X(D) of the set V (D) as follows: 1 ar X iv :1 40 5. 03 26 v1 [ m at h. C O ] 2 M ay 2 01 4 2 On Simultaneous 2-locally-balanced 2-partition for Two Forests with Same Vertices X(D) ≡ {Si(u)/1 ≤ i ≤ exD(v1(D)), u ∈ Si−1, Si(u) 6= ∅} ∪ {S0}. In the further we shall assume, that the consideration of any tree D is automatically implies the choice of the vertex v1(D). Let G is a forest, and D1, D2, ..., D k(G) are its connected components. Define a family of subsets X(G) of the set V (G) as follows: X(G) ≡ k(G) ∪ i=1 X(Di). Let G1and G2 are two forests with V (G1) = V (G2) ≡ V . Define a bipartite multigraph H(G1, G2) = (V1(H(G1, G2)), V2(H(G1, G2)), E(H(G1, G2))) as follows: V1(H(G1, G2)) = X(G1), V2(H(G1, G2)) = X(G2), E(H(G1, G2)) = ∪ v∈V {(u,w)v/u ∈ V1(H(G1, G2)), w ∈ V2(H(G1, G2)), v ∈ u ∩ w}, where E(H(G1, G2)) is understood as multiset containing different elements like (u,w)v1 and (u,w)v2 with v1 6= v2 in a case |u ∩ w| > 1. It is not hard to see that for ∀v ∈ V |{(u,w)v/u ∈ V1(H(G1, G2)), w ∈ V2(H(G1, G2)), v ∈ u ∩ w}| = 1. Taking into account that G1 and G2 are forests we can conclude from the construction of the multigraph H(G1, G2) that there exists an one-to-one correspondence ξ : V → E(H(G1, G2)). From the results of [4] it follows that there exists a 2-partition φ of the set E(H(G1, G2)), at which for ∀v ∈ V1(H(G1, G2)) ∪ V2(H(G1, G2)) bφ(ηH(G1,G2)(v)) ≤ 1. Theorem: If G1and G2 are forests with V (G1) = V (G2) ≡ V , then there exists a simultaneous 2-locally-balanced 2-partition of G1and G2. Proof: Define a 2-partition F of the set V as follows: for ∀v ∈ V F (v) ≡ φ(ξ(v)). We shall be convinced that F is a simultaneous 2-locally-balanced 2-partition of the forests G1and G2. From the construction of the sets X(G1) and X(G2) it follows that for ∀v ∈ V ∃A(v) ∈ X(G1) and ∃B(v) ∈ X(G2) such that |γG1(v)\A(v)| ≤ 1 and |γG2(v)\B(v)| ≤ 1. Therefore it follows that bF (γG1(v)) ≤ bF (A(v)) + 1 = bφ({e ∈ E(H(G1, G2))/ξ(e) ∈ A(v)}) + 1 = bφ(ηH(G1,G2)(v)) + 1 ≤ 2. Similarly, bF (γG2(v)) ≤ 2. Theorem is proved. In the end we bring an example, which explains that not for arbitrary two forests G1and G2 with V (G1) = V (G2) ≡ V there exists a 2-partition f of the set V , which is a simultaneous k-locally-balanced 2-partition of the forests G1and G2 for k ≤ 1. Example: Define trees G1and G2 as follows: G1 = ({v1, v2, v3, v4, v5}, {(v1, v2), (v2, v3), (v3, v4), (v4, v5)}), G2 = ({v1, v2, v3, v4, v5}, {(v1, v5), (v5, v4), (v3, v2), (v2, v1)}). H.G. Tananyan, R.R. Kamalian, 3 Let’s assume that there exists a 2-partition f of the set {v1, v2, v3, v4, v5}, which is a simultaneous k-locally-balanced 2-partition of the trees G1and G2 for k ≤ 1. Without restriction of a generality we can suppose, that f(v1) = 0. From γG1(v2) = {v1, v3} and γG1(v4) = {v3, v5} we can conclude that f(v3) = 1 and f(v5) = 0 . Hence, from γG2(v5) = {v1, v4} and γG2(v1) = {v2, v5} we can conclude that f(v4) = 1 and f(v2) = 1. But it means that bf (γG1(v3)) = 2, which contradicts the property of f .