On b-coloring line, middle and total graph of tadpole graph

We investigate the generalized line graph, total graph and middle graph of the given graphs. First, replace a given graph G with a multigraph M having the same point set as G, then it constructs the total graph T(G) from G. Two suitable subgraphs Mx(G), Lx(G), of Tx(G) are considered as extensions of middle graph M(G) and line graph L(G) respectively.We also consider the connectivities of graphs which are obtained as a special case of Lx(G), Mx(G) and Tx(G). Then we get some extensions of established theorems.

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

The two fastest-growing subfields of graph theory are graph coloringand topological indices. Graph coloring is assigning the colors/values to theedges/vertices or both. A proper coloring of the graph G is assigning colors/valuesto the vertices/edges or both so that no two adjacent vertices/edges share the samecolor/value. Recently, studies involving Chromatic Topological indices that dealtwith different graph coloring were studied. In such studies, the vertex degrees getreplaced with the colors, and the computation is carried out based on the topologicalindex of our choice. We focus on b-Chromatic Zagreb indices and b-Chromaticirregularity indices in this work. This paper discusses the b-Chromatic Zagreb indicesand b-Chromatic irregularity indices of the gear graph, star graph, and itsderived graphs such as the line and middle graph.

Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number chi _d(G) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the \emph{nested complexity length}, which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large `half-graphs' do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large complete multipartite graphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

For a simple connected graph G, let c : V(G) → ℕ be a vertex coloring of G, where adjacent vertices may be colored the same. The neighborhood color set of a vertex v, denoted by NC(v), is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) ≠ NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χs(G). In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

Graph labeling is the assignment of integers to edges or vertices or both depending on some factors which is consequential in the field of graph theory. In this paper, anti-magic graph labeling of two graphs, middle graph and total graph of path graph have been mainly focused. Modified proofs of theorems on anti-magic graph labeling have been introduced here. Both of the theorems are applicable for strong anti-magic graph. The correctness of the proofs has been demonstrated through various examples.

H-trees generalizes the existing notions of trees, higher dimensional trees and k-ctrees. The characterizations and properties of both Pk-trees for k at least 4 and Cn-trees for n at least 5 and their hamiltonian property, dominations, planarity, chromatic and b-chromatic numbers are established. The conditions under which Pk-trees for k at least 3 (resp. Cn-trees for n at least 4), are the line graphs are determined. The relationship between path-trees and split graphs are developed.

In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K4 has * vertex-disjoint triangles and the vertex independence number *. Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are * respectively.

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

We prove that every h-perfect line graph and every t-perfect claw-free graph G has the integer round-up property for the chromatic number: for every non-negative integral weight function c on the vertices of G, the weighted chromatic number of (G, c) can be obtained by rounding up its fractional relaxation. As a corollary, we obtain that the weighted chromatic number can be computed in polynomial-time for these graphs. Finally, we show a new example of a graph operation which preserves the integer round-up property for the chromatic number, and use it to provide a first example of a t-perfect 3-colorable graph which does not have this property.

The research of my dissertation was motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Matthews and Sumner that every 4-connected claw-free graph is hamiltonian. Towards the hamiltonian line graph problem, we proved that every 3-edge-connected, essentially 4-edge-connected graph G has a spanning eulerian subgraph, if for every pair of adjacent vertices u and v, dG(u) + dG(v) ≥ 9. A straight forward corollary is that every 4-connected, essentially 6-connected line graph with minimum degree at least 7 is hamiltonian.;We also investigate graphs G such that the line graph L(G) is hamiltonian connected when L( G) is 4-connected. Ryjacek and Vrana recently further conjectured that every 4-connected line graph is hamiltonian-connected. In 2001, Kriesell proved that every 4-connected line graph of a claw free graph is hamiltonian connected. Recently, Lai et al showed that every 4-connected line graph of a quasi claw free graph is hamiltonian connected, and that every 4-connected line graph of an almost claw free graph is hamiltonian connected. In 2009, Broersma and Vumer discovered the P3-dominating (P3D) graphs as a superfamily that properly contains all quasi claw free graphs, and in particular, all claw-free graphs. Here we prove that every 4-connected line graph of a P3D graph is hamiltonian connected, which extends several former results in this area.;R. Gould [15] asked what natural graph properties of G and H are sufficient to imply that the product of G and H is hamiltonian. We first investigate the sufficient and necessary conditions for G x H being hamiltonian or traceable when G is a hamiltonian graph and H is a tree. Then we further investigate sufficient and necessary conditions for G x H being hamiltonian connected, or edge-pancyclic, or pan-connected.;The problem of group colorings of graphs is also investigated in this dissertation. Group coloring was first introduced by Jeager et al. [21]. They introduced a concept of group connectivity as a generalization of nowhere-zero flows. They also introduced group coloring as a dual concept to group connectivity. Prior research on group chromatic number was restricted to simple graphs, and considered only Abelian groups in the definition of chi g(G). The behavior of group coloring for multigraphs is different to that of simple graphs. Thus we extend the definition of group coloring by considering general groups (both Abelian groups and non-Abelian groups), and investigate the properties of chig for multigraphs

This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal graphs, clique perfect graphs, total graphs, split graphs, line graphs, and dually chordal graphs. We give equations to compute the {k}-clique, minus clique, signed clique, and k-fold clique transversal numbers for suns, and show that the {k}-clique transversal problem is polynomial-time solvable for graphs whose clique transversal numbers equal their clique independence numbers. We also show the relationship between the signed and generalization clique problems and present NP-completeness results for the considered problems on k-trees with unbounded k, planar graphs, doubly chordal graphs, total graphs, split graphs, line graphs, and dually chordal graphs.

Objectives: To find the rainbow vertex connection number for Line, Central, Middle and Total graphs of Comb Graph. Methods/Statistical Analysis: The methods to find the rainbow vertex connection number of any graph G is quite different from other coloring problems. Findings: The rainbow vertex connection number for line graph and middle graph of comb graph      2 n is rainbow vertex connection number for Central graph of Comb graph is 2 and Total graph of Comb graph is     3 n if n=7, 11; is     3 n -1 if n=4k+11, k ≥ 1. While finding the achromatic number for any graph G , noting that no two adjacent vertices receives same color but in the case of rainbow vertex coloring we can assign same color to the adjacent vertices. Application/Improvements: The applications of rainbow vertex connection number are same as rainbow connection number.

Boxicity of line graphs