TY - JOUR
A2 - Krieg, Aloys
AU - Chantasartrassmee, Avapa
AU - Punnim, Narong
PY - 2011
DA - 2011/06/28
TI - An Intermediate Value Theorem for the Arboricities
SP - 947151
VL - 2011
AB - Let G be a graph. The vertex (edge) arboricity of G denoted by a(G) (a1(G)) is the minimum number of subsets into which the vertex (edge) set of G can be partitioned so that each subset induces an acyclic subgraph. Let d be a graphical sequence and let ℛ(d) be the class of realizations of d. We prove that if π∈{a,a1}, then there exist integers x(π) and y(π) such that d has a realization G with π(G)=z if and only if z is an integer satisfying x(π)≤z≤y(π). Thus, for an arbitrary graphical sequence d and π∈{a,a1}, the two invariants x(π)=min(π,d):=min{π(G):G∈ℛ(d)} and y(π)=max(π,d):=max{π(G):G∈ℛ(d)} naturally arise and hence π(d):={π(G):G∈ℛ(d)}={z∈ℤ:x(π)≤z≤y(π)}. We write d=rn:=(r,r,…,r) for the degree sequence of an r-regular graph of order n. We prove that a1(rn)={⌈(r+1)/2⌉}. We consider the corresponding extremal problem on vertex arboricity and obtain min(a,rn) in all situations and max(a,rn) for all n≥2r+2.
SN - 0161-1712
UR - https://doi.org/10.1155/2011/947151
DO - 10.1155/2011/947151
JF - International Journal of Mathematics and Mathematical Sciences
PB - Hindawi Publishing Corporation
KW -
ER -