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.
In this paper among some other results and by using the existance of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph KG(n,2).
Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we study the total dominator chromatic number of some specific 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.
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 $χ_d^t(G)$ of $G$ is the minimum number of color classes in a total dominator coloring of it. In [Total dominator chromatic number of a graph, submitted] the author initialed to study this number in graphs and obtained some important results. Here, we continue it in Mycieleskian graphs. We show that the total dominator chromatic number of the Mycieleskian of a graph $G$ belongs to between $χ_d^t(G)+1$ and $…
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between total dominator chromatic number and chromatic number and total domination number.
Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the total dominator chromatic number of $k$-subdivision of $G$.
We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such a coloring. We show that this notion generalizes the dominator and total dominator chromatic numbers, and provide some general bounds and algorithmic results. We also investigate the specific cases where $P$ is that the subset contains at least one edge or that the subset is connected.
Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we examine the effects on $χ_d^t(G)$ when $G$ is modified by operations on vertex and edge of $G$.
Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In …
We introduce the total dominator edge chromatic number of a graph $G$. A total dominator edge coloring (briefly TDE-coloring) of $G$ is a proper edge coloring of $G$ in which each edge of the graph is adjacent to every edge of some color class. The total dominator edge chromatic number (briefly TDEC-number) $χ'^t_d(G)$ of $G$ is the minimum number of color classes in a TDE-coloring of $G$. We obtain some properties of $χ'^t_d(G)$ and compute this parameter for specific graphs. We examine the effects on $χ'^t_d(G)$ when $G$ is modified by operations on vertex and edge of $G$. Finally, we consid…
Let $G$ be a simple graph of order $n$. A majority dominator coloring of a graph $G$ is proper coloring in which each vertex of the graph dominates at least half of one color class. The majority dominator chromatic number $χ_{md}(G)$ is the minimum number of color classes in a majority dominator coloring of $G$. In this paper we study properties of the majority dominator coloring of a graph. We obtain tight upper and lower bounds in terms of chromatic number, dominator chromatic number, maximum degree, domination and independence number. We also study majority dominator coloring number of sele…
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 comp…
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chro…
In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and anti-arborescences as well as bounds on other orientations of oft studied tree topologies including generalized stars and caterpillars.
For a graph $G = (V(G), E(G))$, a dominating set $D$ is a vertex subset of $V(G)$ in which every vertex of $V(G) \setminus D$ is adjacent to a vertex in $D$. The domination number of $G$ is the minimum cardinality of a dominating set of $G$ and is denoted by $γ(G)$. A coloring of $G$ is a partition $C = (V_{1}, ... ,V_{k})$ such that each of $V_{i}$ in an independent set. The chromatic number is the smallest $k$ among all colorings $C = (V_{1}, ... ,V_{k})$ of $G$ and is denoted by $χ(G)$. A coloring $C = (V_{1}, ... ,V_{k})$ is said to be dominator if, for all $V_{i}$, every vertex $v \in V_{…
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.
This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover that there are infinitely many counterexamples of a graph and subgraph pair for which the subgraph has a larger dominator chromatic number than the larger graph into which it embeds. Finally, a new graph invariant measuring the difference between the dominator chromatic number of a graph and the chromatic number of that graph is established and studied. The …
The circulant graph $C_n(S)$ with connection set $S\subseteq \{1,2,\cdots,n\}$ is the graph with vertex set $V=\{1,\ldots, n\}$ and two vertices $x,y$ are adjacent if $|x-y|\in S$. In this paper, we will calculate the total dominator chromatic number of the circulant graph $C_n(\{a,b\})$ when $n\geq 6$, $gcd(a,n)=1$ and $ a^{-1}b\equiv 3 \pmod{n}$.
A wheel is a graph consisting of an induced cycle of length at least four and a single additional vertex with at least three neighbours on the cycle. We prove that no Burling graph contains an induced wheel. Burling graphs are triangle-free and have arbitrarily large chromatic number, so this answers a question of Trotignon and disproves a conjecture of Scott and Seymour.
Let $G=(V,E)$ be a graph with no isolated vertices. A vertex $v$ totally dominate a vertex $w$ ($w \ne v$), if $v$ is adjacent to $w$. A set $D \subseteq V$ called a total dominating set of $G$ if every vertex $v\in V$ is totally dominated by some vertex in $D$. The minimum cardinality of a total dominating set is the total domination number of $G$ and is denoted by $γ_t(G)$. A total dominator coloring of graph $G$ is a proper coloring of vertices of $G$, so that each vertex totally dominates some color class. The total dominator chromatic number $χ_{td}(G)$ of $G$ is the least number of color…
A domination coloring of a graph $G$ is a proper vertex coloring of $G$ such that each vertex of $G$ dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all domination colorings is called the domination chromatic number, denoted by $χ_{dd}(G)$. In this paper, we study the complexity of this problem by proving its NP-Completeness for arbitrary graphs, and give general bounds and characterizations on several classes of graphs. We also show the relation between dominator chromatic number $χ_{d}(G)$, dominated chromatic n…
Kostochka and Yancey proved that every $4$-critical graph $G$ has $e(G) \geq \frac{5v(G) - 2}{3}$, and that equality holds if and only if $G$ is $4$-Ore. We show that a question of Postle and Smith-Roberge implies that every $4$-critical graph with no $(7,2)$-circular-colouring has $e(G) \geq \frac{27v(G) -20}{15}$. We prove that every $4$-critical graph with no $(7,2)$-colouring has $e(G) \geq \frac{17v(G)}{10}$ unless $G$ is isomorphic to $K_{4}$ or the wheel on six vertices. We also show that if the Gallai Tree of a $4$-critical graph with no $(7,2)$-colouring has every component isomorphic…
A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.
The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$. The least value $k$ for which $G$ admits such labeling is called \emph{additive chromatic number} of $G$. This problem was first presented by Czerwiński, Grytczuk and Żelazny who also proposed a conjecture that for every graph $G$, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete …
In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique numbers onto which wheels and fans fold. We present an interpolation theorem which generalizes their theorems 4.2 and 5.2. We show that their formula for wheels is wrong. We show that for threshold graphs, the achromatic number and folding number coincides with the chromatic number.
A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle. In this paper we show that every subgraph of a wheel graph has list coupled chromatic number at most 5, and this coloring can be found in linear time. We further show that `5' is tight for every wheel graph with at least 5 vertices, and briefly discuss possible generalizations to planar graphs of treewidth 3.
In this paper a subset of High-Dimensional Random Apollonian networks, that we called Wheel Random Apollonian Graphs (WRAG), is considered. We show how to generate a Wheel Random Apollonian Graph from a wheel graph. We analyse some basic graph properties like vertices and edges cardinality, some question concerning cycles and the chromaticity in such type of graphs, we suggest further work on this type of graphs.
A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.
For a graph $G$, $χ(G)$ $(ω(G))$ denote its chromatic (clique) number. A $P_5$ is the chordless path on five vertices, and a $4$-$wheel$ is the graph consisting of a chordless cycle on four vertices $C_4$ plus an additional vertex adjacent to all the vertices of the $C_4$. In this paper, we show that every ($P_5$, $4$-wheel)-free graph $G$ satisfies $χ(G)\leq \frac{3}{2}ω(G)$. Moreover, this bound is almost tight. That is, there is a class of ($P_5$, $4$-wheel)-free graphs $\cal L$ such that every graph $H\in \cal L$ satisfies $χ(H)\geq\frac{10}{7}ω(H)$. This generalizes/improves several previ…
The exact Turán number of the even wheel $W_{2k+2}$ among non-$3$-partite graphs
School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. We say that a multigraph $G$ is locally irregular colorable if it admits a locally irregular edge coloring and we denote by ${\rm lir}(G)$ the locally irregular chromatic index of $G$, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph $G$. We conjecture that for every connected …
On the Turán number of odd-ballooning of $3$-chromatic graphs
School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；School of Mathematics and Finance, Chuzhou University（滁州学院数学与金融学院）；School of Finance Business, Shanghai Normal University（上海师范大学商学院）
We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in the definition of $W({G},q)$ resulting from the fact that at certain special points $q_s$, the following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then present exact calculations of $W({G},q)$ and determine the corresponding analytic structure in the complex $q$ plane for a number of families of graphs ${G}$, i…
It is an open question whether the chromatic number of $t$-perfect graphs is bounded by a constant. The largest known value for this parameter is 4, and the only example of a 4-critical $t$-perfect graph, due to Laurent and Seymour, is the complement of the line graph of the prism $Π$ (a graph is 4-critical if it has chromatic number 4 and all its proper induced subgraphs are 3-colorable). In this paper, we show a new example of a 4-critical $t$-perfect graph: the complement of the line graph of the 5-wheel $W_5$. Furthermore, we prove that these two examples are in fact the only 4-critical $t…
The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h(G) \ge χ(G)$, where $χ(G)$ denotes the chromatic number of $G$. Let $α(G)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h(G) \ge χ(G)$ for all $H$-free graphs $G$ with $α(G) \le 2$, where $H$ is any graph on four vertices with $α(H) \le 2$, $H=C_5$, or $H$ is a particular gra…
Edge-spectral Turán theorems for color-critical graphs with applications
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（极值组合与概率组（ECOPRO），基础科学研究院（IBS））；Department of Mathematics, Stanford University（斯坦福大学数学系）
