English

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring $c$ of a graph $G$ is said to be locally identifying, if for any adjacent vertices $u$ and $v$ with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of $u$ and $v$, respectively, are distinct. Let $\chi_{\rm{lid}}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of $G$. In this paper, we give several bounds on $\chi_{\rm{lid}}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{\rm{lid}}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.

In a directed graph with the unique path property, there is a unique directed path of a given length between any ordered pair of its vertices. Such graphs, their paths, cycles, factors, and other properties, were extensively studied during the years, mainly in the context of de Bruijn graphs, forming the most important family of such graphs. Such a graph is equivalent to a binary matrix for which , where is the all‐ones matrix. In this paper, some new results on the structural properties of such graphs, which are not necessarily de Bruijn graphs, are proved. We mainly concentrate on such graphs for which each vertex has in‐degree two and out‐degree two. The minimum number and the maximum number of vertices which are reached from a given vertex using all paths of length smaller than are considered. We provide an exact formula for the number of factors in these graphs and show that this number depends only on the number of alternating cycles in the graph, where in an alternating cycle every two consecutive edges have opposite directions in the cycle. Constructions of such graphs using several methods are presented. In particular, we present a new construction which provides a large set of such graphs with long alternating cycles. We define the integral graph and the derivative graph and implement these definitions on this family of graphs. Various properties from the point of view of the de Bruijn graph and the theory of nonsingular shift‐registers sequences are also considered. Finally, generalizations for graphs in which the in‐degree and the out‐degree of each vertex are greater than two are also discussed.

The Tur\'{a}n number for a graph $H$ is the least possible number of edges on an $n$-vertex graph with no copy of $H$ as a subgraph. For graphs $G$ and $H$, the \emph{anti-Ramsey number}, denoted $\ar(G,H)$, is the minimum number of colors $d$ such that for any edge coloring with $d$ colors there exists a rainbow copy of $H$. The concept of anti-Ramsey theory was first introduced by Erd\H{o}s, Simonovits, and S\'{o}s in 1973 as a dual problem to traditional Ramsey theory. This thesis investigates Tur\'{a}n theory and anti-Ramsey theory for specific graph families, including cliques in multipartite graphs and matchings in hypergraphs. In Chapter 2, the anti-Ramsey number for disjoint cliques in complete, balanced multipartite graphs is given for certain sizes of cliques and multipartite graphs. In particular, the value $\ar(K_{r\times n}, nK_r)$ is given for all integers $n,r\ge 3$ and $\ar(K_{n,n,n}, 3K_r)$ is given for all integers $n,r\ge 3$. Chapter 3 generalizes Tur\'{a}n and anti-Ramsey theory to the context of hypergraphs. The Tur\'{a}n number for $3$-matchings in complete $3$-uniform hypergraphs is determined. Next, two proofs of the value for the anti-Ramsey number for $4$-matchings in complete hypergraphs are presented. One relies on the Tur\'{a}n number for $3$-matchings, and the other utilizes a method called progressive induction. Chapter 4 features additional work related to combinatorics more generally.

Maximum Laplacian energy among threshold graphs

Acyclic and k-distance coloring of the grid

The classical Nelson — Erdős — Hadwiger problem, which was proposed in the late 1940’s (see [1], [2]), consists in finding or at least estimating the so-called chromatic number χ(ℝ d ) of the Euclidean space ℝ d , where the value χ(ℝ d ) is defined as the minimum number of colours needed to paint all the points in ℝ d in such a way that any two points at the distance 1 apart receive different colours. Of course, in this definition, distance 1 may be replaced by an arbitrary fixed distance a > 0, due to the homogeneity of the Euclidean space. In other words, the quantity χ(ℝ d ) is the usual chromatic number of a graph \(\mathfrak{G}_a^d = \left( {{\mathfrak{G}^d},\mathfrak{E}_a^d} \right)\), provided $${\mathfrak{V}^d} = {\mathbb{R}^d},\mathfrak{E}_a^d = \{ (a,y) \in {\mathfrak{V}^d} \times {\mathfrak{V}^d}:|a - y| = a\} ,\,a > 0$$ .

On f-colorings of nearly bipartite graphs

On the chromatic index of multigraphs and a conjecture of Seymour (I)

Odd graph and its applications to the strong edge coloring

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

The chromatic number of a graph G, denoted χ(G) is the minimum number of colors needed to color vertices of G so that no two adjacent vertices share the same color. A functigraph over a given graph is obtained as follows: Let G' be a disjoint copy of a given G and f be a function f : V(G) → V(G'). The functigraph over G, denoted by C(G, f), is the graph with V(C(G, f)) = V(G) ∪ V(G') and E(C(G, f)) = E(G) ∪ E(G') ∪ {uv : u ∈ V(G), v ∈ V(G'), v = f(u)}. Recently, Chen et al. proved that [Formula: see text]. In this paper, we first provide sufficient conditions on functions f to reach the lower bound for any graph. We then study the attainability of the chromatic numbers of functigraphs. Finally, we extend the definition of a functigraph in different ways and then investigate the bounds of chromatic numbers of such graphs.

A tree in an edge-colored connected graph G is called a rainbow tree if no two edges of it are assigned the same color. For a vertex subset $$S\subseteq V(G)$$ , a tree is called an S-tree if it connects S in G. A k-rainbow coloring of G is an edge-coloring of G having the property that for every set S of k vertices of G, there exists a rainbow S-tree in G. The minimum number of colors that are needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by $${\mathrm {rx}}_k(G)$$ . The Steiner distance d(S) of a set S of vertices of G is the minimum size of an S-tree T. The k-Steiner diameter $${\mathrm {sdiam}}_k(G)$$ of G is defined as the maximum Steiner distance of S among all sets S with k vertices of G. In this paper, we focus on the 3-rainbow index of graphs and find all finite families $$\mathcal {F}$$ of connected graphs, for which there is a constant $$C_\mathcal {F}$$ such that, for every connected $$\mathcal {F}$$ -free graph G, $${\mathrm {rx}}_3(G)\le {\mathrm {sdiam}}_3(G)+C_\mathcal {F}$$ .

coloring takes its name from the map coloring application, we assign labels to vertices. When the numerical value of the labels is unimportant, we call them colors to indicate that they may be elements of any set. In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths. The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this paper we show that the rainbow connection number of an interval graph, which are of the form the rainbow connection number is less than or equal to the diameter of the graph G

Graph colouring is a complex computational problem that focuses on colouring all vertices of a given graph using a minimum number of colours. However, adjacent vertices are restricted from receiving the same colour. Over recent decades, various algorithms have been proposed and implemented to solve such a problem. An interesting algorithm is the hybrid colouring algorithm, which was developed in 1999 by Galinier and Hao. The hybrid colouring algorithm was widely regarded at the time as one of the best performing algorithms for graph colouring. Nowadays, high-performance out-of-order multi-cores have emerged; consequently, executing applications faster and efficiently. Thus, the objective of this paper is to analyse whether the hybrid colouring algorithm can take advantage of multi-core architectures, in terms of performance, or not. For this purpose, we propose and implement a parallel version of the hybrid colouring algorithm that takes advantage of all hardware resources. Several experiments were performed on a machine with two Intel(R) Xeon(R) CPU E5-2630 processors, thus having a total of 24 cores. The experiment proved that the parallel hybrid colouring algorithm, using multi-core architectures, is a significant improvement over the original because it achieves enhancements of up to 40% in terms of the distance to the best chromatic number found in the literature. The expected contribution of this paper is to encourage developers to take advantage of high performance out-of-order multi-cores to solve complex computational problems.

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ?14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.