Vertex arboricity of triangle-free graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order

An Intermediate Value Theorem for the Arboricities

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

An Intermediate Value Theorem for the Arboricities

Let G be a graph. The vertex edge arboricity of G denoted by a G a 1 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, a 1 }, 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, a 1 }, the two invariants x π min π, d : We write d r n : r, r, . . . , r for the degree sequence of an r-regular graph of order n. We prove that a 1 r n { r 1 /2 }. We consider the corresponding extremal problem on vertex arboricity and obtain min a, r n in all situations and max a, r n for all n ≥ 2r 2