An acyclic edge-coloring of a graph G is a proper edge-coloring of G such that the subgraph induced by any two color classes is acyclic. The acyclic chromatic index, χa(G), is the smallest number of colors allowing an acyclic edge-coloring of G. Clearly χa(G) ≥ ∆(G) for every graph G. Cohen, Havet, and Müller conjectured that there exists a constant M such that every planar graph with ∆(G) ≥M h...

The kth chromatic number χk(G) of a graph G is the minimum number of colours needed so that each vertex can be assigned a set of k colours in such a way that colour sets assigned to adjacent vertices are disjoint. Given a graph G = (V,E) and an integerm ≥ 0, them-cone ofG, denoted by μm(G), has vertex set (V ×{0, 1, · · · ,m})∪{u} in which u is adjacent to every vertex of V ×{m}, and (x, i)(y, ...

A deBruijn sequence of order k, or a k-deBruijn sequence, over an alphabet A is a sequence of length |A| in which the last element is considered adjacent to the first and every possible k-tuple from A appears exactly once as a string of k-consecutive elements in the sequence. We will say that a cyclic sequence is deBruijn-like if for some k, each of the consecutive k-element substrings is disti...

The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges $uv$. Previous work has determined EDCN paths, cycles, and spider graphs with three legs. In this paper, we determine petal two petals loop, cycles one chord, fou...

the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

An acyclic coloring of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color and every cycle in G has vertices of at least three different colors. An acyclic k-coloring of G is an acyclic coloring of G with at most k colors. It is NP-complete to decide whether a planar graph G with maximum degree four admits an acyclic 3-coloring [1]...

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arbp(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular c...

A labeling of a graph G is distinguishing if it is only preserved by the trivial automorphism of G. The distinguishing chromatic number of G is the smallest integer k such that G has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product Kk Kn is determined for all k and n. In most of the cases it is equal to the...

An acyclic edge colouring of a graph is a proper edge colouring 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 colouring using k colours and it is denoted by a′(G). Here, we obtain tight estimates on a′(G) for nontrivial subclasses of the family of 2-degenerate graphs. Specifically, we obtain values of...