A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.
Containers have grown into the most dependable and lightweight virtualization platform for delivering cloud services, offering flexible sorting, portability, and scalability. In cloud container services, planner components play a critical role. This enhances cloud resource workloads and diversity performance while lowering costs. We present hybrid optimum and deep learning approach for dynamic scalable task scheduling (DSTS) in container cloud environment in this research. To expand containers virtual resources, we first offer a modified multi-swarm coyote optimization (MMCO) method, which improves customer service level agreements. Then, to assure priority-based scheduling, we create a modified pigeon-inspired optimization (MPIO) method for task clustering and a rapid adaptive feedback recurrent neural network (FARNN) for pre-virtual CPU allocation. Meanwhile, the task load monitoring system is built on a deep convolutional neural network (DCNN), which allows for dynamic priority-based scheduling. Finally, the presentation of the planned DSTS methodology will be estimated utilizing various test vectors, and the results will be associated to present state-of-the-art techniques.
A total coloring of a graph [Formula: see text] is an assignment of colors to the elements of the graph [Formula: see text] such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any simple graph [Formula: see text], [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. In this paper, we prove the tight bound of the total coloring conjecture for the three types of corona products (vertex, edge and neighborhood) of graphs.
A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G, denoted by χ ′′ (G), is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, ∆(G) + 1 ≤ χ ′′ (G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G. In this paper, we prove the Behzad and Vizing conjecture for Indu -Bala product graph, Skew and Converse Skew product graph, Cover product graph, Clique cover product graph and Comb product graph.
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