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The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of cycles and paths.
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Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.
A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring of it. Here, we study the total dominator coloring on central graphs by giving some tight bounds for the total dominator chromatic number of the central of a graph, join of two graphs and Nordhaus-Gaddum-like relations. Also we will calculate the total dominator chromatic number of the central of a path, a cycle, a wheel, a complete graph and a complete multipartite graph.
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Delta(G)$, chromatic index $chi(G)$ and total chromatic number $chi(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $chi(G)leq Delta(G)+mu(G) +1$ for a multigraph $G$, where $mu(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $chi(G)=chi(G)$ if $chi(G)geq Delta(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $chi(G)=chi(G)$ if $chi(G)geq max{ Delta(G)+2, |V(G)|+1}$ in this note. Consequently, $chi(G) = chi(G)$ if $chi(G) ge Delta(G) +2$ and $G$ has a spanning edge-chromatic critical subgraph.
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this article, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitely the total dominator chromatic number of the middle graph of several known families of graphs.
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)leq R_k(P_n)leq R_k(C_n)leq kn+o(n)$. The upper bound was recently improved by Sarkozy who showed that $R_k(C_n)leqleft(k-frac{k}{16k^3+1}right)n+o(n)$. Here we show $R_k(C_n) leq (k-frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
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