Let G be a finite graph with p vertices and x its chromatic polynomial. A combinatorial interpretation is given to the positive integer (-l)px(-A), where h is a positive integer, in terms of acyclic orientations of G. In particular, (-l)Px(-1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in ...

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Theref...

A walk W in edge-colored graph is called properly colored (PC) if every pair of consecutive edges in W is of different color. We study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity of type i is a proper superset of graphs of acyclicity of type i+1, i = 1, 2, 3, 4. The first three types are equivalent to the absence of PC cycles, PC trails, and PC walks. Wh...

We consider constrained proper edge colorings of the following type: Given a positive integer j and a family F of connected graphs on 3 or more vertices, we require that the subgraph formed by the union of any j color classes has no copy of any member of F . This generalizes some well-known types of colorings such as acyclic edge colorings, distance-2 edge colorings, low treewidth edge coloring...

Let G = (V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N [v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v ∈ V , there is exactly one d ∈ D dominating v. An edge set M ⊆ E is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph...

The graph in Figure S1 is undirected, since the edges have no orientation. If it is necessary to specify the orientation of edges, then the graph is directed and this is graphically represented by inserting arrows on the edges, having (x, y) to denote a strictly ordered pair, such that x corresponds to the tail and y to the head of the arrow. The degree of a vertex is the number of edges that c...

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set asD and has an edge between x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of...

let $g=(v‎, ‎e)$ be a graph with $p$ vertices and $q$ edges‎. ‎an emph{acyclic‎ ‎graphoidal cover} of $g$ is a collection $psi$ of paths in $g$‎ ‎which are internally-disjoint and cover each edge of the graph‎ ‎exactly once‎. ‎let $f‎: ‎vrightarrow {1‎, ‎2‎, ‎ldots‎, ‎p}$ be a bijective‎ ‎labeling of the vertices of $g$‎. ‎let $uparrow!g_f$ be the‎ ‎directed graph obtained by orienting the...

Directed acyclic graphs have been used fruitfully to represent causal structures (Pearl 1988). However, in the social sciences and elsewhere models are often used which correspond both causally and statistically to directed graphs with directed cycles (Spirtes 1995). Pearl (1993) discussed predicting the effects of intervention in models of this kind, so-called linear non-recursive structural e...