An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index χa(G) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that χa(G) ≤ ∆(G) + 2 for any simple graph G with maximum degree ∆(G). In this paper, we prove that every planar graph G admits an acyclic edg...

Vertex coloring of a graph is the assignment of labels to the vertices of the graph so that adjacent vertices have different labels. In the case of polyhedral graphs, the chromatic number is 2, 3, or 4. Edge coloring problem and face coloring problem can be converted to vertex coloring problem for appropriate polyhedral graphs. We have been developed an interactive learning system of polyhedra,...

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

Let G be a graph with chromatic number (G) and maximum degree (G). A star colouring of G is a function that assigns a colour to each vertex such that adjacent vertices receive distinct colours, and there is no bichromatic 4-vertex path. The star chromatic number st(G) is the minimum number of colours in a star colouring of G. Star colourings of subdivisions of graphs are investigated. Let G0 be...

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...

We give upper bounds for the generalised acyclic chromatic number and generalised acyclic edge chromatic number of graphs with maximum degree d, as a function of d. We also produce examples of graphs where these bounds are of the correct order.

An (m,n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m,n)colored mixed graph G to an (m,n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is an arc (edge) of color c in H . The (m,n)-colored mixed chromatic number χ(m,n)(G) of...

A proper edge colouring of a graph is said to be acyclic if every cycle of G receives at least three colors. The acyclic chromatic index, denoted , is the least number of colors required for an acyclic edge color of . This paper shows an upper bound of the acyclic chromatic index of a class of graphs which can be expressed as the Cartesian product of some graphs. We also give exact values for s...

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...