Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V , let c(v) denote the sum of the weight of v ∈ V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) 6= c(v) for any two adjacent vertices u and v of G. We show that for every 2-connected plane graph there is such a proper vertex co...

Abstract: A particle moves among the vertices of an (m + 1)-gon which are labeled clockwise as 0, 1, . . . , m. The particle starts at 0 and thereafter at each step it moves to the adjacent vertex, going clockwise with a known probability p, or counterclockwise with probability 1 − p. The directions of successive movements are independent. What is the expected number of moves needed to visit al...

A simple random walk on a graph is defined in which a particle moves from one vertex to any adjacent vertex, each with equal probability. The expected hitting time is the expected number of steps to get from one vertex to another before returning to the starting vertex. In this paper, using the electrical network approach, we provide explicit formulae for expected hitting times for simple rando...

We consider the class ER(n, d, λ) of edge-regular graphs for some n > d > λ, i.e., graphs regular of degree d on n vertices, with each pair of adjacent vertices having λ common neighbors. It has previously been shown that for such graphs with λ > 0 we have n ≥ 3(d − λ) and much has been done to characterize such graphs when equality holds. Here we show that n ≥ 3(d − λ) + 1 if λ > 0 and d is od...

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that v...

As a natural extension of previously defined graph labelings, we introduce in this paper a new magic labeling whose evaluation is based on the neighbourhood of a vertex. We define a 1-vertex-magic vertex labeling of a graph with v vertices as a bijection f taking the vertices to the integers 1, 2, . . . , v with the property that there is a constant k such that at any vertex x, ∑ y∈N(x) f(y) = ...

Zsolt Tuza y Latest update : September 8, 1997 Abstract We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment. Mostly, the list colorings and the precoloring extensions are considered. In one of th...

A method is proposed for obtaining the local clique set from a neighbourhood system. The Markov random field model, which is used extensively in image processing, is defined with respect to a neighbourhood system. The mathematical interpretation of the model is defined with respect to the corresponding clique set. We present a systematic method for extracting the complete local clique set from ...

A gambler moves between the vertices 1, . . . , n of a graph using the probability distribution p1, . . . , pn. Multiple cops pursue the gambler on the graph, only being able to move between adjacent vertices. We investigate the expected capture time for the gambler against k cops as a function of n and k for three versions of the game: • known gambler: the cops know the gambler’s distribution ...