A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number...

A 3-consecutive C-coloring of a graph G = (V,E) is a mapping φ:V → ℕ such that every path on three vertices has at most two colors. We prove general estimates on the maximum number ${\left(\chi ̅\right)}_{3CC}\left(G\right)$ of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with ${\left(\chi ̅\right)}_{3CC}\left(G\right)\ge k$ for k = 3 and k = 4.

In this paper we study the existence of unavoidable paths on three vertices in sparse graphs. A path uvw on three vertices u, v, and w is of type (i, j, k) if the degree of u (respectively v, w) is at most i (respectively j, k). We prove that every graph with minimum degree at least 2 and average degree strictly less than m contains a path of one of the types [...] Moreover, no parameter of this description can be improved.

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is 3-transitive if the existence of the directed path (u,v,w,x) of length 3 in D implies the existence of the arc (u,x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g., to characterize 3-transitive digraphs that are transitive and to characterize 3-transitive digraphs with...

A $(0,2)$-graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0,\lambda )$-graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0,2)$-graph. $(0,2)$-graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0,\lambda )$-graphs and more specifically in $(0,2)$-graphs,...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is transitive if for every three distinct vertices u, v,w ∈ V (D), (u, v), (v,w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈ A(D). A very useful structural characterization of transitive digraphs has been known for a long time, and...

It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48

Decomposable (probabilistic) models are log-linear models generated by acyclic hypergraphs, and a number of nice properties enjoyed by them are known. In many applications the following selection problem naturally arises: given a probability distribution $p$ over a finite set $V$ of $n$ discrete variables and a positive integer $k$, find a decomposable model with tree-width $k$ that best fits $p$. If $\mathscr{H}$ is the generating hypergraph of a decomposable model and $p_{\mathscr{H}}$ is the estimate of $p$ under the model, we can measure...

A probabilistic communication structure considers the setting with communication restrictions in which each pair of players has a probability to communicate directly. In this paper, we consider a more general framework, called a probabilistic communication structure with fuzzy coalition, that allows any player to have a participation degree to cooperate within a coalition. A maximal product spanning tree, indicating a way of the greatest possibility to communicate among the players, is introduced...

We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [6] (Shue, Anderson, Dey 1998). We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one.

Let $G$ be a graph with vertex set $V\left(G\right)$, and let $k\ge 1$ be an integer. A subset $D\subseteq V\left(G\right)$ is called a $k$-dominating set if every vertex $v\in V\left(G\right)-D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma }_k\left(G\right)$ of $G$ is the minimum cardinality of a $k$-dominating set in $G$. If $G$ is a graph with minimum degree $\delta \left(G\right)\ge k+1$, then we prove that $${\gamma }_{k+1}\left(G\right)\le \frac{\left|V\left(G\right)\right|+{\gamma }_k\left(G\right)}{2}.$$ In addition, we present a characterization of a special class of graphs attaining equality in this inequality.

Given graphs A, B and C for which A×C ≅ B×C, it is not generally true that A ≅ B. However, it is known that A×C ≅ B×C implies A ≅ B provided that C is non-bipartite, or that there are homomorphisms from A and B to C. This note proves an additional cancellation property. We show that if B and C are bipartite, then A×C ≅ B×C implies A ≅ B if and only if no component of B admits an involution that interchanges its partite sets.