On the Total Set Chromatic Number of Graphs

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χs(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T(G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs

On the Total Set Chromatic Number of Graphs

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χs(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings, that is, we study set colorings of the total graph of graphs. Given a graph G = (V; E); its total graph T (G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First; we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs

The Set Chromatic Numbers of the Middle Graph of Graphs

For a simple connected graph G; let c : V (G) → N be a vertex coloring of G; where adjacent vertices may be colored the same. The neighborhood color set of a vertex v; denoted by NC(v); is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) neq NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χ_s(G). In this work; the set chromatic number of graphs is studied inrelation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles; paths; star graphs; double-star graphs; and some trees of height 2. Moreover; we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally; we develop an algorithm for constructingan optimal set coloring of the middle graph of trees of height 2 under some assumptions

On the set chromatic number of the join and comb product of graphs

A vertex coloring c : V(G) → of a non-trivial connected graph G is called a set coloring if NC(u) ≠ NC(v) for any pair of adjacent vertices u and v. Here, NC(x) denotes the set of colors assigned to vertices adjacent to x. The set chromatic number of G, denoted by χs (G), is defined as the fewest number of colors needed to construct a set coloring of G. In this paper, we study the set chromatic number in relation to two graph operations: join and comb prdocut. We determine the set chromatic number of wheels and the join of a bipartite graph and a cycle, the join of two cycles, the join of a complete graph and a bipartite graph, and the join of two bipartite graphs. Moreover, we determine the set chromatic number of the comb product of a complete graph with paths, cycles, and large star graphs