Beck’s conjecture and multiplicative lattices

Improvement on asymptotic density of packing families derived from multiplicative lattices

Quantales were introduced by C.J Mulvey (2) to provide a non-commutative extension of the concept of locale (1). Certain authors (4) restrict the term quantale to mean those multiplicative lattices for which each element is right sided (a & a = a). In this paper by a quantale we mean a complete multiplicative lattice with 1 =T. The paper deals with Localic reflections of quantales. A condition for neat elements of a coherent quantale to form a locale has been studied.

We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, including lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space; we further explore continuous maps between lower spaces.

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

Circular chromatic numbers of Mycielski's graphs

In this article, a new idea of fuzzy fractional coloring of fuzzy graph is presented and fuzzy fractional chromatic number is defined. A relationship between fuzzy fractional chromatic number and fuzzy fractional clique number is established. Some properties of fuzzy chromatic number of fuzzy graphs and fuzzy fractional chromatic number of fuzzy graphs are proved and the concept of k-strong adjacent vertices is introduced. Fuzzy chromatic number and fuzzy fractional chromatic number have been calculated on lexicographic product of two fuzzy graphs. Also, fuzzy chromatic number, independence number and fuzzy fractional chromatic number have been investigated on disjoint union of two fuzzy graphs. Lastly, a real life application of fuzzy fractional coloring on fuzzy graph is discussed.

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Some problems on induced subgraphs

The quantum chromatic number of a graph $G$ is sandwiched between its\nchromatic number and its clique number, which are well known NP-hard\nquantities. We restrict our attention to the rank-1 quantum chromatic number\n$\\chi_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is\ndefined under stronger constraints. We study its relation with the chromatic\nnumber $\\chi(G)$ and the minimum dimension of orthogonal representations\n$\\xi(G)$. It is known that $\\xi(G) \\leq \\chi_q^{(1)}(G) \\leq \\chi(G)$. We\nanswer three open questions about these relations: we give a necessary and\nsufficient condition to have $\\xi(G) = \\chi_q^{(1)}(G)$, we exhibit a class of\ngraphs such that $\\xi(G) < \\chi_q^{(1)}(G)$, and we give a necessary and\nsufficient condition to have $\\chi_q^{(1)}(G) < \\chi(G)$. Our main tools are\nKochen-Specker sets, collections of vectors with a traditionally important role\nin the study of noncontextuality of physical theories, and more recently in the\nquantification of quantum zero-error capacities. Finally, as a corollary of our\nresults and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum\nchromatic number, we give a family of Kochen-Specker sets of growing dimension.\n

Let G = (V, E) be a graph with vertex set V and edge set E. Given non negative integers r, s and t, an [r, s, t]-coloring of a graph G is a proper total coloring where the neighboring elements of G (vertices and edges) receive colors with a certain difference r between colors of adjacent vertices, a difference s between colors of adjacent edges and a difference t between colors of a vertex and an incident edge. Thus [r, s, t]-colorings generalize the classical colorings of graphs and can have applications in different fields like scheduling, channel assignment problem, etc. The [r, s, t]-chromatic number ? r,s,t (G) of G is the minimum k such that G admits an [r, s, t]-coloring. In our paper we propose several bounds for the [r, s, t]-chromatic number of the cartesian and direct products of some graphs.

Zero-divisor graph of a poset with respect to an automorphism

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.

Graph theory can give a representation of abstract mathematical systems such as groups or rings. We have many graph representations for a group, in this study we use the coprime graph representation for a generalized quaternion group to find the numerical invariants of the graph, which are the clique number and the chromatic number. The main results obtained from this study are the clique number of the coprime graph representation for the generalized quaternion group is equal to the chromatic number of the coprime graph representation for the generalized quaternion group for each case of the order.

Relations of vertex distinguishing total chromatic numbers between a subgraph and its supergraph