On b-vertex and b-edge critical graphs

A b-coloring of a graph G is a proper coloring of G with k colors such that each color class has a vertex that is adjacent to at least one vertex of every other color classes. The b-chromatic number is the largest integer k for which G has a b-coloring with k colors. In this paper, we present some results on b-coloring in complementary prism graphs.

On [formula omitted]-coloring of powers of hypercubes

The b-chromatic number of powers of hypercube

On the [formula omitted]-coloring of [formula omitted

The [formula omitted]-chromatic number of regular graphs via the edge-connectivity

Tree graph is a connected graph and has no circuits. Tree graphs used in this study include: broom graph, firework graph, banana tree graph, centipede graph, E graph, and double star graph. Graph coloring is the process of giving color to graph elements with the rule that neighboring elements must not have the same color and the number of colors used must be as minimal as possible. b-Coloring of a graph G is a coloring of the vertices of G such that each color class has at least one vertex adjacent to all other color classes. The b-Chromatic number of a graph G is denoted by φ (G), is the largest integer k such that G has a b-coloring with k colors. The limit of b-coloring of graph G with maximum degree ∆(G) is as follows. χ(G)≤φ(G)≤∆(G)+1.χ(G) is the chromatic number of a graph G where χ(G) is the minimum value of the color required for proper coloring of graph G. While ∆(G) is the maximum degree of the vertices in graph G. This study uses an exploratory research type with axiomatic deductive method and pattern detection method. Based on the results of this study, the results of the b-coloring analysis on the tree graph family are known. The results of this study are expected to be used as study material and development of scientific knowledge related to b-coloring of other graphs

b-Chromatic Sum of Mycielskian of Paths

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

A b-colouring of a graph G is a proper colouring of the vertices of G such that there exists a vertex in each colour class joined to at least one vertex in each other colour classes. The b-chromatic number of a graph G, denoted by φ(G) is the largest integer k such that G has a b-colouring with k colours. The Mycielskian or Mycielski graph µ(H) of a graph H with vertex set {v1, v2, , vn} is a graph G obtained from H by adding n + 1 new vertices {u, u1, u2, , un}, joining u to each vertex ui (1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, we obtain the b-chromatic number of Mycielskian of paths, complete graphs, complete bipartite graphs and wheels.

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With Pelikánová and Pokorná, we showed that this conjecture is true if each color class has size ${\Omega}(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size ${\Omega}(k)$, which improved the previous result and is only a constant factor away from Aharoni's conjecture. We also consider what happens when the condition on the number of colors is relaxed.

On Aharoni’s rainbow generalization of the Caccetta–Häggkvist conjecture

The b-chromatic index of graphs

Large dense neighbourhoods and turán's theorem

We show that the following fundamental edge-colouring problem can be solved in polynomial time for any given constant B: given a simple graph G, find an edge-colouring of G where each colour is assigned to at most B edges and which, subject to this condition, has the fewest number of colour classes.

Supereulerian width of dense graphs