On the partial Grundy coloring of graphs

A coloring of a graph $G=(V,E)$ is a partition $\{V_1, V_2, \ldots, V_k\}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every $j<i$. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number $\Gamma(G)$ of a graph $G$ is the maximum number of colors in a Grundy coloring. We provide two new upper bounds on Grundy number of a graph and a stronger version of the well-known Nordhaus-Gaddum theorem. In addition, we give a new characterization for a $\{P_{4}, C_4\}$-free graph by supporting a conjecture of Zaker, which says that $\Gamma(G)\geq \delta(G)+1$ for any $C_4$-free graph $G$.

This work is a logical continuation of previous works related to the study of promising topologies for networks-on-chip based on the direct and root product of circulant graphs. In this article, we describe and compare the static characteristics of the hierarchical method of constructing more complex structures compared to the root and direct products of graphs. Also in this paper, the result of the analysis of the integration of atomic structures (the smallest structural units of scalable topologies) into various levels of hierarchical structure is presented. A comparison of the obtained structures with the previously studied structures, which are the results of the application of the root product of Paley graphs, is presented.

This article analyzes new promising topological solutions for the on-chip communication subsystem for network-on-chip (NoCs). A study of Paley graphs as a subclass of circulant graphs is given. The main parameters (diameter, average path length, graph density, number of edges and degrees of vertices) of modifications of Paley graphs are studied using the direct product of Paley graphs with Paley graphs and ring graph. Other types of products of graphs are considered, namely strong, tensor and lexicographic. Due to the more preferable characteristics, the direct product of the graphs was chosen as the most suitable among the considered ones. A comparative analysis of obtained graphs is provided.

Let [Formula: see text] be a connected undirected graph. A vertex coloring [Formula: see text] of [Formula: see text] is an [Formula: see text]-vertex coloring if for each vertex [Formula: see text] in [Formula: see text], the number of different colors assigned to [Formula: see text] is at most [Formula: see text]. The [Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is the maximum number of colors which are used in an [Formula: see text]-vertex coloring of [Formula: see text]. In this paper, we provide sharp bounds for [Formula: see text] of a graph [Formula: see text] in terms of its vertex cover number, maximum degree and diameter, respectively. We also determine precise values for [Formula: see text] in some cases.

Complexity of coloring graphs without paths and cycles

The problem of on-line coloring of an arbitrary graphs is known to be hard. Here we consider the problem of on-line coloring in the simplified situation where the input graph is KK s,t -free. We show that the on-line coloring algorithm with the First Fit strategy of proposed by Capponi and Pilotto in [1] is the best one for KK 1,t -free graphs (t?3). A.Capponi and C.Pilotto have shown that this algorithm achieves a competitive ratio of t?1 while we show that it is the best possible. However for the family of KK s,t -free graphs (s?2, t?2) there exists no on-line coloring algorithm with a competitive function. The problem of an on-line cliques covering for these families is hard. There exists no on-line cliques covering algorithm with a competitive function for the family of KK s,t -free graphs (s? 1, t?3). The additional assumption that the input graph is given in a connected way does not help a lot and does not change our results described above.

Estimation of Laplacian spectra of direct and strong product graphs

A graph is $k$-total colourable if there is an assignment of $k$ different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product graphs are determined. In particular, a sufficient condition is given for direct products of bipartite graphs to have total chromatic number equal to its maximum degree plus one. Partial results towards the total chromatic number of $K_n\times K_m$ are also established.

Indicated coloring of a graph G is a coloring in which there are two players Ann and Ben, Ann picks a vertex and Ben chooses a color for this vertex. The aim of Ann is to achieve a proper coloring of the whole graph G, while Ben tries to block the same. The smallest number of colors required for Ann to win the game on a graph G is called the indicated chromatic number of G and is denoted by \(\chi _i(G).\) In this paper, we prove that \(T\Box C_n, T\Box K_{n_1,n_2,\dots ,n_m}\) and \(K_{n_1,n_2,\dots ,n_m}\Box C_m\) are k-indicated colorable for all k greater than or equal to the indicated chromatic number of their corresponding Cartesian product, where T is any tree. Also we prove that \(\chi _i(K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}) =\chi (K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}).\) Finally we have given non-trivial examples of graphs G and H for which \(\chi _i(G\Box H) >\chi (G\Box H).\)

Maximum Face-Constrained Coloring of Plane Graphs

Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round, the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben’s strategy) is called the indicated chromatic number of G, denoted by [Formula: see text]. In this paper, we examine whether the Mycielskian of G, [Formula: see text], is k-indicated colorable for all [Formula: see text], whenever G is l-indicated colorable for all [Formula: see text]. In this direction, we prove that the Mycielskian of the bipartite graphs, complete multipartite graphs, [Formula: see text]-free graphs, [Formula: see text]-free graphs, [Formula: see text]-free graphs and [Formula: see text]-free graphs are k-indicated colorable for all k greater than or equal to the indicated chromatic number of the corresponding graphs.

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.

On weighted efficient total domination

Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal rank preserving direct product, and the normal product. Counter-examples are given for the strong product as well as the non-rank-preserving and the maximal rank preserving direct product.

Given a k-coloring S1, . . . , Sk of G, a vertex in Si is said to be greedy if it has neighbors in Sj, for every j &lt; i. Thus, a Grundy coloring can be seen as a coloring where every vertex is greedy. In contrast, a partial Grundy coloring is defined as a coloring in which each color class has at least one greedy vertex; the maximum number of colors in a partial Grundy coloring is denoted by ∂Γ(G). In this paper, we investigate some conjectures about Grundy colorings, already known not to hold, adapted to partial Grundy colorings. We prove that, while two of them also do not hold, a third does. Namely, that given a graph G and a matching M of G, we have that ∂Γ(G) ≤ 2∂Γ(G − M).