We show that every connected induced subgraph of a graph $G$ is dominated by an induced connected split graph if and only if $G$ is $\cal{C}$-free, where $\cal{C}$ is a set of six graphs which includes $P_7$ and $C_7$, and each containing an induced $P_5$. A similar characterisation is shown for the class of graphs which are dominated by induced complete split graphs. Motivated by these results, we study structural descriptions of some classes of $\cal{C}$-free graphs. In particular, we give structural descriptions for the class of ($P_7$,$C_7$,$C_4$,gem)-free graphs and for the class of ($P_7$,$C_7$,$C_4$,diamond)-free graphs. Using these results, we show that every ($P_7$,$C_7$,$C_4$,gem)-free graph $G$ satisfies $χ(G) \leq 2ω(G)-1$, and that every ($P_7$,$C_7$,$C_4$,diamond)-free graph $H$ satisfies $χ(H) \leq ω(H)+1$. These two upper bounds are tight for any subgraph of the Petersen graph containing a $C_5$.
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…
For a simple graph $G$, a domination coloring of $G$ is a proper vertex coloring such that every vertex of $G$ dominates at least one color class, and every color class is dominated by at least one vertex. The domination chromatic number, denoted by $χ_{dd}(G)$, is minimum number of colors among all domination colorings of $G$. In this paper, we discuss the effects of some typical operations on $χ_{dd}(G)$, such as vertex (edge) removal, vertex (edge) contraction, edge subdivision, and cycle extending.
We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.
An edge coloring of a tournament $T$ with colors $1,2,\dots,k$ is called \it $k$-transitive \rm if the digraph $T(i)$ defined by the edges of color $i$ is transitively oriented for each $1\le i \le k$. We explore a conjecture of the second author: For each positive integer $k$ there exists a (least) $p(k)$ such that every $k$-transitive tournament has a dominating set of at most $p(k)$ vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erd\H os, Sands, Sauer and Woodrow (so the conjecture is interestin…
A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together …
Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.
Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set $U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex $v\in \cup_{i\notin S} A_i$ there is a w in U such that (w,v) is in E(D). Let beta(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h=h(beta(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colo…
The game domination number is a graph invariant that arises from a game, which is related to graph domination in a similar way as the game chromatic number is related to graph coloring. In this paper we show that verifying whether the game domination number of a graph is bounded by a given integer is PSPACE-complete. This contrasts the situation of the game coloring problem whose complexity is still unknown.
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_{…
More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of…
Using a relation between domination number and incidence chromatic number, we obtain necessary and sufficient conditions for $r$-regular graphs to be $(r+1)$-incidence colorable. Also, we determine the optimal Nordhaus-Gaddum inequality for the incidence chromatic number.
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…
This paper continues the results of "Domination Cover Pebbling: Graph Families." An almost sharp bound for the domination cover pebbling (DCP) number for graphs G with specified diameter has been computed. For graphs of diameter two, a bound for the ratio between the cover pebbling number of G and the DCP number of G has been computed. A variant of domination cover pebbling, called subversion DCP is introducted, and preliminary results are discussed.
In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph $G$, called the sub-$k$-domination number and denoted $sub_k(G)$. We show that $sub_k(G)$ is a computationally efficient sharp lower bound on the $k$-domination number of $G$, and improves on several known lower bounds. We also characterize the sub-$k$-domination numbers of several families of graphs, provide structural results on sub-$k$-domination, and explore properties of graphs which are $sub_k(G)$-critical with respect to addition and deletion of vertices and edges.
A totally silver coloring of a graph G is a k--coloring of G such that for every vertex v \in V(G), each color appears exactly once on N[v], the closed neighborhood of v. A totally silver graph is a graph which admits a totally silver coloring. Totally silver coloring are directly related to other areas of graph theory such as distance coloring and domination. In this work, we present several constructive characterizations of totally silver graphs and bipartite totally silver graphs. We give several infinite families of totally silver graphs. We also give cubic totally silver graphs of girth u…
We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős-Rényi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.
Domination polynomial of co-maximal graphs of integer modulo ring
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；School of Mechanical Engineering, Shandong University of Technology（山东理工大学机械工程学院）
Given a graph property $\mathcal{P}$, F. Harary introduced in 1985 $\mathcal{P}$-colorings, graph colorings where each colorclass induces a graph in $\mathcal{P}$. Let $χ_{\mathcal{P}}(G;k)$ counts the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. It turns out that $χ_{\mathcal{P}}(G;k)$ is a polynomial in $\mathbb{Z}[k]$ for each graph $G$. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial $χ(G;k)$. We show that the characteristic…
A \emph{domatic} (\emph{total domatic}) \emph{$k$-coloring} of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that each vertex contains vertices of all $k$ colors in its closed neighborhood (neighborhood). The \emph{domatic} (\emph{total domatic}) \emph{number} of $G$, denoted $d(G)$ ($d_t (G)$), is the maximum $k$ for which $G$ has a domatic (total domatic) $k$-coloring. In this paper, we show that for two non-trivial graphs $G$ and $H$, the domatic and total domatic numbers of their Cartesian product $G \cart H$ is bounded above by $\max\{|V(G)|, |V(H)|\}$ and below b…
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.
A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Therefore, two open problems presented by Fabrici and Madaras [The structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are solved. Furthermore, we prove that each 1-planar graph $G$ with …
An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiamčík (1978) conjectured that $\chiup_{a}'(G) \leq Δ(G) + 2$, where $Δ(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $κ$ is {\em $κ$-deletion-minimal} if $\ch…
Paired Disjunctive Domination Number of Middle Graphs
Maritime Faculty Dokuz Eylul University（德内兹大学海事学院）；Computer Programing Department Izmir Konak Vocational School（伊兹密尔科纳克职业学院计算机编程系）；Department of Mathematics Ege University（艾登大学数学系）
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 clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with $q$ colors. For fixed $q \ge 2$, we give an $\mathcal{O}^{\star}(q^{tw})$-time algorithm when the input graph is given together with one of its tree decompositions of width $tw$. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unb…
For a graph $G$, let $χ(G)$ and $ω(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies $χ(G)\le \lceil\frac{5ω(G)}{4}\rceil$. Moreover, this bound is best possible.
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the $(p,1)$-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the $(p,1)$-total labelling number of every 1-planar graph $G$ is at most $Δ(G)+2p-2$ provided that $Δ(G)\…
A power dominating set of a graph is a set of vertices that observes every vertex in the graph by combining classical domination with an iterative propagation process arising from electrical circuit theory. In this paper, we study the power domination polynomial of a graph $G$ of order $n$, defined as $\mathcal{P}(G;x)=\sum_{i=1}^n p(G;i) x^i$, where $p(G;i)$ is the number of power dominating sets of $G$ of size $i$. We relate the power domination polynomial to other graph polynomials, present structural and extremal results about its roots and coefficients, and identify some graph parameters …
The upper (mixed) domination Ramsey number $u(m, n)$($v(m,n)$) is the smallest integer $p$ such that every $2$-coloring of the edges of $K_p$ with color red and blue, $Γ(B) \geq m$ or $Γ(R) \geq n$ ($β(R) \geq n$); where $B$ and $R$ is the subgraph of $K_p$ induced by blue and red edges, respectively; $Γ(G)$ is the maximum cardinality of a minimal dominating set of a graph $G$. First, we prove that $v(3,n)=t(3,n)$ where $t(m,n)$ is the mixed irredundant Ramsey number i.e. the smallest integer $p$ such that in every two-coloring $(R, B)$ of the edges of $K_p$, $IR(B) \geq m$ or $β(R) \geq n$ ($…
The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-ha…
A 3-coloring of the elements of an abelian group is said to be rainbow--free if there is no 3-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow--free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow-free 3-coloring of cyclic groups.
A graph $G$ is said to be $k$-$γ_{c}$-critical if the connected domination number $γ_{c}(G)$ is equal to $k$ and $γ_{c}(G + uv) < k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $ζ$ be the number of cut vertices of $G$ and let $ζ_{0}$ be the maximum number of cut vertices that can be contained in one block. For an integer $\ell \geq 0$, a graph $G$ is $\ell$-factor critical if $G - S$ has a perfect matching for any subset $S$ of vertices of size $\ell$. It was proved that, for $k \geq 3$, every $k$-$γ_{c}$-critical graph has at most $k - 2$ cut vertices and the graphs with max…
We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $χ_3$. We bound $χ_3$ by other known structural parameters. We also present two classes of g…
We generalize the family of $(σ, ρ)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as $k$-trapezoid graphs, Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, $k$-polygon graphs,…
