In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G)→ 2[α], and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contain...

An acyclic coloring of a graph G is a coloring of its vertices such that : (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an a...

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D′(G). We prove that D′(G) 6 D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chrom...

We provide a characterization of several graph parameters (the acyclic chromatic number, the arrangeability, and a sequence of parameters related to the expansion of a graph) in terms of forbidden subdivisions. Let us start with several definitions. Throughout the paper, we consider only simple undirected graphs. A graph G = sdt(G) is the t-subdivision of a graph G, if G is obtained from G by r...

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C(u) be the set of colors of vertex u and edges incident to u under f . For an IE-total coloring f of G using k colors, if C(u) 6= C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-color...

An acyclic coloring of a graph G is a coloring of its vertices such that:(i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. Recently it has been proved that any graph of maximum degree 5 has an acyclic chromatic number at most 8. In this paper we prese...

Acyclic-coloring of a graph G = (V; E) is a partitioning of V , such that the induced subgraph of each partition is acyclic. The minimum number of such partitions of V is deened as the vertex arboricity of G. An O(n) algorithm (n = jV j) for acyclic-coloring of planar graphs with 3 colors is presented. Next, an O(n 2) heuristic is proposed which produces a valid acyclic-2-coloring of a planar g...