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Equitable total coloring of corona of cubic graphs  [PDF]
Hanna Furmańczyk
Computer Science , 2015,
Abstract: The minimum number of total independent sets of $V \cup E$ of graph $G(V,E)$ is called the \emph{total chromatic number} of $G$, denoted by $\chi"(G)$. If difference of cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of $V \cup E$ is called the \emph{equitable total chromatic number}, and denoted by $\chi"_=(G)$. In this paper we consider equitable total coloring of corona of cubic graphs, $G \circ H$. It turns out that, independly on equitable total chromatic numbers of $G$ and $H$, equitable total chromatic number of corona $G \circ H$ is equal to $\Delta(G \circ H) +1$. Thereby, we confirm TCC and ETCC conjectures for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.
The Locating Chromatic Number of the Join of Graphs  [PDF]
Ali Behtoei
Mathematics , 2011,
Abstract: Let $f$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(V_1,V_2,...,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be the ordered $k$-tuple $c_{{}_\Pi}(v):=(d(v,V_1),d(v,V_2),...,d(v,V_k)),$ where $d(v,V_i)=\min\{d(v,x)|x\in V_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $f$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic number of the join of graphs. We show that when $G_1$ and $G_2$ are two connected graphs with diameter at most two, then $\Cchi_{{}_L}(G_1+G_2)=\Cchi_{{}_L}(G_1)+\Cchi_{{}_L}(G_2)$, where $G_1+G_2$ is the join of $G_1$ and $G_2$. Also, we determine the locating chromatic numbers of the join of paths, cycles and complete multipartite graphs.
Equitable chromatic threshold of Kronecker products of complete graphs  [PDF]
Zhidan Yan,Wei Wang
Mathematics , 2012,
Abstract: A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic threshold of a graph $G$, denoted by $\chi_=^*(G)$, is the minimum $k$ such that $G$ is equitably $k^\prime$-colorable for all $k^\prime \ge k$. Let $G\times H$ denote the direct product of graphs $G$ and $H$. For $n\ge m\ge 2$ we prove that $\chi_=^*(K_{m} \times K_n)$ equals $\lceil\frac{mn}{m+1}\rceil$ if $n\equiv 2,...,m (\textup{mod} m+1)$, and equals $m\lceil\frac{n}{s^\star}\rceil$ if $n\equiv 0,1 (\textup{mod} m+1)$, where $s^\star$ is the minimum positive integer such that $s^\star \nmid n$ and $s^\star\ge m+2.$
Equitable chromatic threshold of complete multipartite graphs  [PDF]
Zhidan Yan,Wei Wang
Mathematics , 2012,
Abstract: A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $\chi_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k\ge t$. We develop a formula and a linear-time algorithm which compute the equitable chromatic threshold of an arbitrary complete multipartite graph.
Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs  [PDF]
Boris Brimkov,Illya V. Hicks
Computer Science , 2015,
Abstract: 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.
On r-equitable chromatic threshold of Kronecker products of complete graphs  [PDF]
Wei Wang,Zhidan Yan,Xin Zhang
Computer Science , 2013,
Abstract: A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most $r$. The $r$-equitable chromatic threshold of a graph $G$, denoted by $\chi_{r=}^*(G)$, is the minimum $k$ such that $G$ is $r$-equitably $k'$-colorable for all $k'\ge k$. Let $G\times H$ denote the Kronecker product of graphs $G$ and $H$. In this paper, we completely determine the exact value of $\chi_{r=}^*(K_m\times K_n)$ for general $m,n$ and $r$. As a consequence, we show that for $r\ge 2$, if $n\ge \frac{1}{r-1}(m+r)(m+2r-1)$ then $K_m\times K_n$ and its spanning supergraph $K_{m(n)}$ have the same $r$-equitable colorability, and in particular $\chi_{r=}^*(K_m\times K_n)=\chi_{r=}^*(K_{m(n)})$, where $K_{m(n)}$ is the complete $m$-partite graph with $n$ vertices in each part.
On the Set of Circular Total Chromatic Numbers of Graphs  [PDF]
Mohammad Ghebleh
Mathematics , 2011, DOI: 10.1016/j.disc.2012.10.023
Abstract: For every integer $r\ge3$ and every $\eps>0$ we construct a graph with maximum degree $r-1$ whose circular total chromatic number is in the interval $(r,r+\eps)$. This proves that (i) every integer $r\ge3$ is an accumulation point of the set of circular total chromatic numbers of graphs, and (ii) for every $\Delta\ge2$, the set of circular total chromatic numbers of graphs with maximum degree $\Delta$ is infinite. All these results hold for the set of circular total chromatic numbers of bipartite graphs as well.
Total dominator chromatic number and Mycieleskian graphs  [PDF]
Adel P. Kazemi
Mathematics , 2013,
Abstract: 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 $\chi_d^t(G)$ 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 $\chi_d^t(G)+1$ and $\chi_d^t(G)+2$, and then characterize the family of graphs the their total dominator chromatic numbers are each of them.
Total dominator chromatic number of specific graphs  [PDF]
Saeid Alikhani,Nima Ghanbari
Mathematics , 2015,
Abstract: 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 $\chi_d^t(G)$ 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.
Equitable Colorings of Corona Multiproducts of Graphs  [PDF]
Hanna Furma?czyk,Marek Kubale,Vahan V. Mkrtchyan
Computer Science , 2012,
Abstract: A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $\chi_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In "Equitable colorings of Cartesian products of graphs" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of corona products of graphs. In particular, we obtain some results regarding the equitable chromatic number for $l$-corona product $G \circ ^l H$, where $G$ is an equitably 3- or 4-colorable graph and $H$ is an $r$-partite graph, a path, a cycle or a complete graph. Our proofs are constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of $G$. Moreover, we confirm Equitable Coloring Conjecture for corona products of such graphs. This paper extends our results from \cite{hf}.
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