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Register a new userTotal dominator coloring of central graphs
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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.
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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.
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.
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.
Hadwigers conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwigers conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)sqcup E(G)$ with $c_1,c_2in V(G)sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwigers conjecture is true for $T(G)$. The total chromatic number $chi(G)$ of a graph $G$ is defined to be equal to the chromatic number of its total graph. That is, $chi(G)=chi(T(G))$. Another well-known conjecture in graph theory, the total coloring conjecture or TCC, states that for every graph $G$, $chi(G)leqDelta(G)+2$, where $Delta(G)$ is the maximum degree of $G$. We show that if a weaker version of the total coloring conjecture (weak TCC) namely, $chi(G)leqDelta(G)+3$, is true for a class of graphs $mathcal{F}$ that is closed under the operation of taking subgraphs, then Hadwigers conjecture is true for the class of total graphs of graphs in $mathcal{F}$. This motivated us to look for classes of graphs that satisfy weak TCC. It may be noted that a complete proof of TCC for even 4-colorable graphs (in fact even for planar graphs) has remained elusive even after decades of effort; but weak TCC can be proved easily for 4-colorable graphs. We noticed that in spite of the interest in studying $chi(G)$ in terms of $chi(G)$ right from the initial days, weak TCC is not proven to be true for $k$-colorable graphs even for $k=5$. In the second half of the paper, we make a contribution to the literature on total coloring by proving that $chi(G)leqDelta(G)+3$ for every 5-colorable graph $G$.
For a given graph $G$, the least integer $kgeq 2$ such that for every Abelian group $mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)rightarrow mathcal{G}$ so that $sum_{xin N(u)}f(xu) eq sum_{xin N(v)}f(xv)$ for each edge $uvin E(G)$ is called the textit{group twin chromatic index} of $G$ and denoted by $chi_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $chi_g(G)leq Delta(G)+3$ for all graphs without isolated edges, where $Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $chi_g(G)leq 2(Delta(G)+{rm col}(G))-5$, where ${rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $mathbb{Z}_k$.
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