A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. The b-spectrum Sb(G) of a graph G is the set of positive integers k...

Given a graph G, b coloring seeks proper of G with the descriptive property that every color class in contains dominating vertex. We have to minimize number colors coloring. In this paper, we show how chromatic and b-chromatic changes operators relation between has been explored.

In this paper we show upper bounds for the sum and the product of the lower domination parameters and the chromatic index of a graph. We also present some families of graphs for which these upper bounds are achieved. Next, we give a lower bound for the sum of the upper domination parameters and the chromatic index. This lower bound is a function of the number of vertices of a graph and a new gr...

The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded treeor pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel...

We introduce H -join decompositions of graphs, indexed by a fixed bipartite graph H . These decompositions are based on a graph operation that we call H -join, which adds edges between two given graphs by taking partitions of their two vertex sets, identifying the classes of the partitions with vertices of H , and connecting classes by the pattern H . H -join decompositions are related to modul...

The b-chromatic number (G) of a graphG is defined as the largest number k for which the vertices of G can be colored with k colors satisfying the following property: for each i, 1 i k, there exists a vertex xi of color i such that for all j = i, 1 j k there exists a vertex yj of color j adjacent to xi . A graph G is b-perfect if each induced subgraph H of G has (H) = (H), where (H) is the chrom...

In this paper an improved bound on the chromatic number of Pancake graph $P_n, n\geqslant 2$, is presented. The obtained using a subadditivity property graph. We also investigate equitable coloring $P_n$. An $(n-1)$-coloring based efficient dominating sets given and optimal $4$-colorings are considered for small $n$. It conjectured that $P_n$ coincides with its any $n\geqslant 2$.