Path (or cycle)-trees with Graph Equations involving Line and Split Graphs

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.

The notion of word-representable graphs has been extensively studied. It is well known that the set of word-representable graphs are exactly the graphs whose edges can be ordered in a semi-transitive manner. Thus the set of word-representable graphs is decidable. This paper gives an alternative and simpler proof of decidability of word-representable graphs. The second part of the paper introduces a notion called multi-word-representability. Many classes of graphs - planar graphs, interval graphs, split graphs, co-bipartite graphs and line graphs - are shown to be two word-representable. An upper bound on the number of words needed to represent $$k-$$ colourable graphs has also been calculated.

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

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 $\mathcal{P}_{g,p}$.

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.

Irwing and Manlove instigate the knowledge of b -colorings and provided some inequalities for extreme bounds to explore the parameter on b -coloring. He denoted m-degree m(G) to determine the b -chromatic number. Here the parameter on b-coloring of Line graph of Tadpole Graph, Middle graph and Total graph of tadpole graph are being investigated. Also comparison of parameters on coloring and b-coloring has been done.

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.

Boxicity of line graphs

The line graph of a graph G denoted as L(G) has vertex set E(G) in which two vertices are adjacent if they correspond to adjacent edges in G. The lict graph and litact graph of G, denoted as \(L_c(G)\) and \(L_{ct}(G)\), respectively having vertex set \(E(G)\cup C(G)\) (here C(G) is the set of cut-vertices of G), two of these vertices will be adjacent in \(L_c(G)\) if they correspond to adjacent edges of G or one vertex is an edge e of G and other vertex is a cut-vertex c of G such that e is incident to c; and two vertices in \(L_{ct}(G)\) be adjacent if they are adjacent or incident elements of G. In this paper, we establish results on cliques and clique chromatic numbers in line, lict and litact graphs of any graph.KeywordsLine graphLict graphLitact graphCliqueClique chromatic numberMSC(2010):Primary: 05C75Secondary: 05C76

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G= (V,E) and a spanning subgraph Hof G(the backbone of G), a i¾?-backbone coloring for Gand His a proper vertex coloring Vi¾?{1,2,...} of Gin which the colors assigned to adjacent vertices in Hdiffer by at least i¾?. The main outcome of earlier studies is that the minimum number i¾? of colors for which such colorings Vi¾?{1,2,...,i¾?} exist in the worst case is a factor times the chromatic number (for all studied types of backbones). We show here that for split graphs and matching or star backbones, i¾? is at most a small additive constant (depending on i¾?) higher than the chromatic number. Despite the fact that split graphs have a nice structure, these results are difficult to prove. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on i¾? than the previously known bounds.

A b-coloring of a graph G is a proper coloring of G with k colors such that each color class has a vertex that is adjacent to at least one vertex of every other color classes. The b-chromatic number is the largest integer k for which G has a b-coloring with k colors. In this paper, we present some results on b-coloring in complementary prism graphs.

In this paper, we introduce chromatic number of line graph using α-cuts. The concept of chromatic number of fuzzy graphs wasintroduced by Munoz et.al, later Eslahchi and onagh. They are defined by the fuzzy chromatic number of complete graphs(kn),cycle graph(cn),star graph(sn),wheel graph(wn),and line graph are found and results are summarized.

In this paper, we utilize the signless Laplacian eigenvalues of line graphs and the dominant number to determine the limits of chromatic numbers for the line graphs of certain special graphs. The application of Vizing’s theorem helps to justify the existence of larger boundaries in this particular case.

Parameterized and exact algorithms for class domination coloring

On [formula omitted]-coloring of powers of hypercubes