We show that the 2-edge-colored chromatic number of a class of simple graphs is bounded if and only if the acyclic chromatic number is bounded for this class. Recently, the CSP dichotomy conjecture has been reduced to the case of 2-edge-colored homomorphism and to the case of 2-vertex-colored homomorphism. Both reductions are rather long and follow the reduction to the case of oriented homomorp...

Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G′ be the graph obtained from G...

The point-distinguishing chromatic index of a graph represents the minimum number of colours in its edge colouring such that each vertex is distinguished by the set of colours of edges incident with it. Asymptotic information on jumps of the point-distinguishing chromatic index of Kn,n is found.

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling   with $d$ labels  that is preserved only by a trivial automorphism. The distinguishing chromatic number $chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. We prove that if $G$ is a bipartite graph of girth at least six with the maximum ...

A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that is adjacent to a vertex of every other color. The b-chromatic number b(G) of a graph G is the largest k such that G admits a b-coloring with k colors. A graph G is b-chromatic edge critical if for any edge e of G, the b-chromatic number of G − e is less than the b-chromatic number of G. We ...

let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ a (G). We prove that χ a (G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges w...

With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishin...

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F), is defined as the maximum a(G) over all the graphs G ∈ F . In this pape...