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This paper investigates the coexistence of two competing species on random geometric graphs (RGGs) in continuous time. The species grow by occupying vacant sites according to Richardson's model, while simultaneously competing for occupied…
We consider two-player games played on finite colored graphs where the goal is the construction of an infinite path with one of the following frequency-related properties: (i) all colors occur with the same asymptotic frequency, (ii) there…
Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to…
We study competition on scale-free random graphs, where the degree distribution satisfies an asymptotic power-law with infinite variance. Our competition process is such that the two types attempt at occupying vertices incident to the…
Coloring is a notoriously hard problem, and even more so in the online setting, where each arriving vertex has to be colored immediately and irrevocably. Already on trees, which are trivially two-colorable, it is impossible to achieve…
One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy…
A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of…
We study a two-species competition model in a patchy advective environment, where the species are subject to both directional drift and undirectional random dispersal between patches and there are losses of individuals in the downstream end…
This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front…
Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In…
In the first part of this paper, we consider weighted domination in the case where the vertices of the complete graph on~\(n\) vertices are equipped with independent and identically distributed (i.i.d.) weights. We use the probabilistic…
Resource competition is a fundamental interaction in natural communities.However little is known about competition in spatial environments where organisms are able to regulate resource distributions. Here, we analyze the competition of two…
A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the…
Suppose that the vertices of ${\mathbb Z}^d$ are assigned random colors via a finitary factor of independent identically distributed (iid) vertex-labels. That is, the color of vertex $v$ is determined by a rule that examines the labels…
In 1970 Hajnal and Szemer\'edi proved a conjecture of Erd\"os that for a graph with maximum degree $\Delta$, there exists an equitable $\Delta+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand…
We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on $N$ vertices. The processes are allowed to spread with different rates, start from vertex subsets of different…
We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to…
A simple but empirically efficient heuristic algorithm for the edge-coloring of graphs is presented. Its basic idea is the displacement of "conflicts" (repeated colors in the edges incident to a vertex) along paths of adjacent vertices…
Consider a graph whose vertices are colored in one of two colors, say black or white. A white vertex is called integrated if it has at least as many black neighbors as white neighbors, and similarly for a black vertex. The coloring as a…
Vizing's theorem states that any graph of maximum degree $\Delta$ can be properly edge colored with at most $\Delta+1$ colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any…