Proper Vertex Research Articles

On the computational complexity of the bipartizing matching problem

Given a graph $$G=(V,E)$$ , an edge bipartization set of G is a subset $$E'\subseteq E(G)$$ such that $$G'=(V,E{\setminus } E')$$ is bipartite. An edge bipartization set that is also a matching of G is called a bipartizing matching of G. Let $${\mathscr {BM}}$$ be the family of all graphs admitting a bipartizing matching. Although every graph has an edge bipartization set, the problem of recognizing graphs G having bipartizing matchings ( $$G\in \mathscr {BM}$$ ) is challenging. A (k, d)-coloring of a graph G is a k-coloring of V(G) such that each vertex of G has at most d neighbors with the same color as itself. Clearly a (k, 0)-coloring is a proper vertex k-coloring of G and, for any $$d>0$$ , the k-coloring is non-proper, also known as defective. A graph belongs to $$\mathscr {BM}$$ if and only if it admits a (2, 1)-coloring. Lovász (1966) proved that for any integer $$k>0$$ , any graph of maximum degree $$\varDelta $$ admits a $$\left( k,\lfloor \varDelta /k \rfloor \right) $$ -coloring. In this paper, we show that it is NP-complete to determine whether a 3-colorable planar graph of maximum degree 4 belongs to $$\mathscr {BM}$$ , i.e., (2, 1)-colorable. Besides, we show that it is NP-complete to determine whether planar graphs of maximum degree 6 or 8 admit a (2, 2) or (2, 3)-coloring, respectively. Due to Lovász (1966), our results are tight in the sense that on graphs with maximum degree three, five, and seven, there always exists a (2, 1), (2, 2), and (2, 3)-coloring, respectively. Finally, we present polynomial-time algorithms for particular graph classes, besides some remarks on the parameterized complexity of the problem of recognizing graphs in $${\mathscr {BM}}$$ .

On the complexity of coloring ‐graphs

AbstractAn ‐graph is a graph that can be partitioned into r independent sets and ℓ cliques. In the Graph Coloring problem, we are given as input a graph G, and the objective is to determine the minimum integer k such that G admits a proper vertex k‐coloring. In this work, we describe a Poly versus NP‐hard dichotomy of this problem regarding the parameters r and ℓ of ‐graphs, which determine the boundaries of the ‐hardness of Graph Coloring for such classes. We also analyze the complexity of the problem on ()‐graphs under the parameterized complexity perspective. We show that given a (2, 1)‐partition of a graph G, finding an optimal coloring of G is ‐complete even when K1 is a maximal clique of size 3; XP but W[1]‐hard when parameterized by ; fixed‐parameter tractable (FPT) and admits a polynomial kernel when parameterized by . Besides, concerning the case where K1 is a maximal clique of size 3, a P versus NPh dichotomy regarding the neighborhood of K1 is provided; furthermore, an FPT algorithm parameterized by the number of vertices having no neighbor in K1 is presented.

The achromatic number of K_{6} □ K_{7} is 18

A vertex colouring \(f:V(G)\to C\) of a graph \(G\) is complete if for any two distinct colours \(c_1, c_2 \in C\) there is an edge \(\{v_1,v_2\}\in E(G)\) such that \(f(v_i)=c_i\), \(i=1,2\). The achromatic number of \(G\) is the maximum number \(\text{achr}(G)\) of colours in a proper complete vertex colouring of \(G\). In the paper it is proved that \(\text{achr}(K_6 \square K_7)=18\). This result finalises the determination of \(\text{achr}(K_6 \square K_q)\).

On \delta^(k)-colouring of Powers of Paths and Cycles

In a proper vertex colouring of a graph, the vertices are coloured in such a way that no two adjacent vertices receive the same colour, whereas in an improper vertex colouring, adjacent vertices are permitted to receive same colours subjected to some conditions. An edge of an improperly coloured graph is said to be a bad edge if its end vertices have the same colour. A near proper colouring is a colouring which minimises the number of bad edges by restricting the number of colour classes that can have adjacency among their own elements. The $\delta^{(k)}$- colouring is a near proper colouring of $G$ consisting of $k$ given colours, which minimises the number of bad edges by permitting at most one colour class to have adjacency among the vertices in it. In this paper, we determine the number of bad edges of powers of Paths $(P_{n})$ and powers of Cycles $(C_{n})$.

On Equitable Coloring of Book Graph Families

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by atmost one. The notion of equitable coloring was introduced by Meyer in 1973. A proper $h-$colorable graph $K$ is said to be equitably h-colorable if the vertex sets of $K$ can be partioned into $h$ independent color classes $V_1, V_2,...,V_h$ such that the condition $\left|\left|V_i\right|-\left|V_j\right|\right| \leq 1$ holds for all different pairs of $i$ and $j$ and the least integer $h$ is known as equitable chromatic number of $K$. In this paper, we find the equitable coloring of book graph, middle, line and central graphs of book graph.

Star chromatic number of Harary graphs

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every path on four vertices in [Formula: see text] is not 2-colored. The star chromatic number is the minimum number of colors required to star color [Formula: see text] and it is denoted by [Formula: see text]. In this paper, the star coloring of Harary graphs [Formula: see text], where [Formula: see text] is even and [Formula: see text] is odd, is discussed.

Bilangan Kromatik Graceful pada Keluarga Graf Unicyclic

Suppose $ G $ is a graph, where $ G = \{V (G), E (G) \} $. Graceful coloring is defined by $ c: V (G) \to \{1,2, ..., k \} $ which induces a proper edge coloring $ c': E (G) \to \{1,2, ..., k- 1 \}$ defined by $c'(xy)=|c(x)-c(y)|$, where $ k \geq 2 $, $ k \in N $. Coloring is said to be graceful if these 3 conditions are satisfied, namely the proper vertex color, the proper edge color, and the edge color, which are the absolute difference between the color of the accident vertex. The subgraph $H$ on that graceful coloring is smaller than the $G$. Furthermore, one of the subgraphs in the unicyclic graph family is a cycle graph. The graceful chromatic number on a graph denoted by $ \chi_g (G) $, is the optimum number of graceful colors from graph $G$. This research aims to find graceful chromatic numbers in the unicyclic graph family, namely bull graphs, net graphs, cricket graphs, caveman graphs, peach graphs, and flowerpot graphs. The results of this study indicate that $\chi_g(C_l) \geq 4$, where $C_l$ is a unicyclic graphs.

Chromatic Zagreb and irregularity polynomials of graphs

Graph coloring is an assignment of colors, labels or weights to elements of a graph subject to certain constraints. Coloring the vertices of a graph in such a way that adjacent vertices are having different colors is called proper vertex coloring. A proper vertex coloring using minimum parameters of colors is studied extensively in recent literature. In this paper, we define new coloring related polynomials, called chromatic Zagreb polynomials and chromatic irregularity polynomials, in terms of minimal parameter coloring and structural characteristics of graphs such as distances and degrees of vertices.

Construction of Blocked Factorial Designs to Estimate Main Effects and Selected Two-Factor Interactions

AbstractTwo-level factorial designs are widely used in industry. For experiments involving n factors, the construction of designs comprising 2n and 2n-p factorials, arranged in blocks of size 2q is investigated. The aim is to estimate all main effects and a selected subset of two-factor interactions. Designs constructed according to minimum aberration criteria are shown to not necessarily be the most appropriate designs in this situation. A design construction approach is proposed which exploits known results on proper vertex colourings in graph theory. Examples are provided to illustrate the results and construction strategies. Particular consideration is given to the special case of designs with blocks of size four and tables of designs are given for this block size.

Acyclic Coloring of Graphs with Maximum Degree 7

The acyclic chromatic number a(G) of a graph G is the minimum number of colors such that G has a proper vertex coloring and no bichromatic cycles. For a graph G with maximum degree $$\Delta $$ , Grunbaum (1973) conjectured $$a(G)\le \Delta +1$$ . Up to now, the conjecture has only been shown for $$\Delta \le 4$$ . In this paper, it is proved that $$a(G)\le 12$$ for $$\Delta =7$$ , thus improving the result $$a(G)\le 17$$ of Dieng et al. (in: Proc. European conference on combinatorics, graph theory and applications, 2010).

Local Super Antimagic Total Labeling for Vertex Coloring of Graphs

Let G=(V,E) be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from V∪E to {1,2,…,|V|+|E|} such that if for each uv∈E(G) then w(u)≠w(v), where w(u)=∑uv∈E(G)f(uv)+f(u). If the range set f satisfies f(V)={1,2,…,|V|}, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color w(v) assigning the vertex v. The local super antimagic total chromatic number of graph G, χlsat(G) is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

Dominated and dominator colorings over (edge) corona and hierarchical products

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 chromatic number of hierarchical products with two factors determined. An application of dominated colorings in genetic networks is also proposed.

On star coloring of modular product of graphs

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$. \par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $V(G)$ and $E(G)$, respectively. Branko Gr\"{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.

Fixed interval scheduling with third‐party machines

AbstractWe study a problem of scheduling n jobs on machines of two types: in‐house machines and third‐party machines. Scheduling on in‐house machines incurs no additional costs, while using third‐party machines implies costs depending on their number and the time of usage. Each job has a fixed time interval for being processed which can be divided and allocated among several machines, as long as there is only one machine processing the job at any time. No machine can process more than one job at a time. Jobs can be rejected, and they are of different importance that is reflected in the weight of each job. The objective is to find a subset of the jobs and the number of third‐party machines for any period of time so that the accepted jobs can be feasibly scheduled, the total weight of the accepted jobs is maximized, and the total machine usage costs does not exceed a given upper bound. We also study a similar problem in which the objective is to maximize the total time at which at least one job is processed. Both problems are encountered in situations in which certain activities with given start and completion times have to be serviced by human operators. Examples are air traffic control and the monitoring safe vehicle unloading. Other examples are the employment of subcontractors in agriculture, construction or transportation. We will present NP‐hardness proofs, polynomial and pseudo‐polynomial optimal algorithms and an approximation algorithm for these problems and their special cases. These problems admit graph‐theoretical interpretations associated with finding independent sets and a proper vertex coloring in interval graphs.

Local antimagic chromatic number of trees - I

Let G = (V, E) be a connected graph with |V|= n and |E|= m. A bijection f : E → {1, 2, … , m} is called a local antimagic lableing if for any two adjacent vertices u and v, w(u) ≠ w(v), where w(u)= ∑ e ϵE(u) f (e) and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number χla (G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we determine the local antimagic chromatic number of several families of trees.

Uniquely list colorability of the graph Kn-2+Om

Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely k-list colorable if there is a list assignment L such that jL(v)j = k for every vertex v and the graph has exactly one L-coloring with these lists. In this paper, we characterize uniquely list colorability of the graph G = Kn2 + Om. We shall prove that if n = 2 then G is uniquely 3-list colorable if and only if m >= 9, if n = 3 and m >=1 then G is uniquely 3-list colorable, if n >=4 then G is uniquely k-list colorable with k =[m/2]+1, and if m>=n-1, entonce G es UnLC.

ON STAR COLORING OF DEGREE SPLITTING OF COMB PRODUCT GRAPHS

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph.

Local super antimagic total vertex coloring of some wheel related graphs

Let G be a simple and finite graph with vertex set V and edge set E. The local antimagic total labeling of G is a map from V ∪ E to the set of positive integers from 1 to |V| + |E| such that the weight of two adjacent vertices are distinct. The weight of a vertex v is calculated by the sum of the label of the vertex v and the labels of all edges incident to it. If the vertices of G is labelled by the smallest labels, that is, {1, 2,…, |V|}, then the such labeling is called local super antimagic total labeling. Thus, the local super antimagic total labeling induces a proper vertex coloring of G where the vertex v is colored by the weight of vertex v. The minimum number of colors taken over all colorings induced by super local antimagic total labeling of G, is called local super antimagic total chromatic number of graph G, and denoted by χlsat (G). In this paper, we consider the local super antimagic total chromatic number of some wheel related graphs such as fans, even gear graphs, and sun flower graphs.

On the local antimagic vertex coloring of sub-devided some special graph

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

The upper bound of vertex local antimagic edge labeling on graph operations

Let G(V(G), E(G)) be a connected simple graph and let u, v be two vertices of graph G. A bijective function f from the edge set of G to the natural number up to the number of edges in G is called a vertex local antimagic edge labeling if for any two adjacent vertices v and v′, w(v) = w(v′), where , and E(v) is the set of vertices which is incident to v. Thus any vertex local antimagic edge labeling induces a proper vertex coloring of G where the vertex v is assigned with the color w(v). The vertex local antimagic chromatic number χla (G) is the minimum number of colors taken over all coloring induced by vertex local antimagic edge labeling of G. In this paper, we study the vertex local antimagic edge labeling of graphs and determine the upper bound of vertex local antimagic chromatic number on graph operation, namely cartesian product of graphs, corona product of graphs, and power graph.