Determining equitable total chromatic number for infinite classes of complete [formula omitted]-partite graphs

The minimum number of total independent partition sets of $V \cup E$ of a graph $G=(V,E)$ is called the \emph{total chromatic number} of $G$, denoted by $\chi''(G)$. If the difference between 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 is denoted by $\chi''_=(G)$. In this paper we consider equitable total coloring of coronas of cubic graphs, $G \circ H$. It turns out that, independly on the values of equitable total chromatic number of factors $G$ and $H$, equitable total chromatic number of corona $G \circ H$ is equal to $\Delta(G \circ H) +1$. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.

A proper total-coloring of graph G is said to be equitable if the number of elements (vertices and edges) in any two color classes differ by at most one, which the required minimum number of colors is called the equitable total chromatic number. In this paper, we prove some theorems on equitable total coloring and derive the equitable total chromatic numbers of Pm Sn, Pm Fn and Pm Wn.

In Math Program 55(1992), 129–168, Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for balanced bipartite graphs whose maximum degree is at most 3. We in fact obtain results for more general classes, namely linear balanceable and subcubic balanceable graphs. Additionally, we prove that cubic balanced graphs contain a pair of twins, a result that was conjectured by Morris, Spiga, and Webb in ( Discrete Math 310(2010), 3228–3235).

We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c > 0 such that for every family F of n x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turan-type results on incomparability graphs of partially ordered sets.

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

A b-colouring of a graph G is a proper colouring of the vertices of G such that there exists a vertex in each colour class joined to at least one vertex in each other colour classes. The b-chromatic number of a graph G, denoted by φ(G) is the largest integer k such that G has a b-colouring with k colours. The Mycielskian or Mycielski graph µ(H) of a graph H with vertex set {v1, v2, , vn} is a graph G obtained from H by adding n + 1 new vertices {u, u1, u2, , un}, joining u to each vertex ui (1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, we obtain the b-chromatic number of Mycielskian of paths, complete graphs, complete bipartite graphs and wheels.

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures. A total dominator coloring (briefly, a TDC) of a graph G is a proper coloring (i.e. a partition of the vertex set V(G) into independent subsets named color classes) in which each vertex of the graph is adjacent to all of vertices of some color class. The total dominator chromatic number of G is the minimum number of color classes in a TDC of G. In this paper among some other results and by using the existence of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph for each

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

We consider the Gathering problem where a swarm of weak robots disposed on the vertices of an anonymous graph are required to meet at one vertex from where they do not move anymore. In our recent work [Cicerone et al., SIROCCO’19], we have shown how synchronicity heavily affects the design of resolution algorithms within the standard Look-Compute-Move (LCM) model. In particular, we have investigated two dense and highly symmetric topologies: complete graphs and complete bipartite graphs. We characterized all solvable configurations for synchronous robots, whereas it is known that in complete graphs asynchronous robots cannot solve the problem, ever. Instead of approaching directly the asynchronous case in complete bipartite graphs, we asked what happens in the so-called semi-synchronous model, that is robots are synchronized but they are not necessarily all active within all LCM cycles. It turns out that still the gathering can never be accomplished on complete graphs, whereas challenging cases arise in complete bipartite graphs. We provide a distributed algorithm solving the problem for a wide set of possible configurations. For most of the remaining ones instead we provide impossibility results and a few of ad hoc resolution algorithms studied for very specific cases. Over all, still a full characterization is missing but our study points out how difficult might be to derive a general argument that catches all peculiarities. Moreover, some of our approaches reveal new insights that might be very useful for the resolution of other tasks.

One of the topics in graph theory that is interesting and developed continuously is metric dimension. It has some new variation concepts, such as star metric dimension. The order set of Z={z1,z2,…,zn }⊆V(G) called star resolving set of connected graph G if Z is Star Graph and for every vertex in G has different representation to the set Z. The representation is expressed as the distance d(u,z), it is the shortest path from vertex u to z for every u,z∈V(G). Star basis of a graph is the smallest cardinality of star resolving set. The number of vertex in star basis is called star metric dimension of G which denoted by Sdim(G). The purpose of this article is to determine the characteristic of star metric dimension and the value of star metric dimension of some classes of graphs. The method which is used in this study is library research. Some of the results of this research are complete graph has Sdim(Kn )=n-1, for n≥3, bipartite graph K(2,n) has Sdim (K(2,n) )=n, for n≥3. Besides complete bipartite graph hasn’t star metric dimension or for m,n≥3 or it can said that Sdim (K(m,n) )=0. Another graph, that is Fan graph has Sdim (Fn )=2 for 2≤n≤5 and for n≥6 Sdim (Fn )=(2n+3)/5.

SIZE RAMSEY NUMBERS INVOLVING MATCHINGS