Related papers: Chip-Firing and Riemann-Roch Theory for Directed G…
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road…
We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness…
A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of…
Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed $k$-ary tree…
In this paper, we study the dynamics of sand grains falling in sand piles. Usually sand piles are characterized by a decreasing integer partition and grain moves are described in terms of transitions between such partitions. We study here…
We show that, in almost every $n$-vertex random directed graph process, a copy of every possible $n$-vertex oriented cycle will appear strictly before a directed Hamilton cycle does, except of course for the directed cycle itself.…
Evolutionary games on graphs play an important role in the study of evolution of cooperation in applied biology. Using rigorous mathematical concepts from a dynamical systems and graph theoretical point of view, we formalize the notions of…
It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results…
Admissible strategies, i.e. those that are not dominated by any other strategy, are a typical rationality notion in game theory. In many classes of games this is justified by results showing that any strategy is admissible or dominated by…
Let $\phi \colon \Gamma_2 \rightarrow \Gamma_1$ be a harmonic morphism of connected graphs. We show that an arithmetical structure on $\Gamma_1$ can be pulled back via $\phi$ to an arithmetical structure on $\Gamma_2$. We then show that…
Riemann surfaces which are set by algebraic, algebroid and inverse functions are considered. A method for describing these Riemann surfaces by graphs is proposed. Each such Riemann surface is assigned to a special type of graph - profile.…
A theorem of Cohn and Lempel [J. Combin. Theory Ser. A 13 (1972), 83-89] gives an equality relating the number of circuits in a directed circuit partition of a 2-in, 2-out digraph to the GF(2)-nullity of an associated matrix. This equality…
We investigate the concept of effective resistance in connection graphs, expanding its traditional application from undirected graphs. We propose a robust definition of effective resistance in connection graphs by focusing on the duality of…
Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph of an equivalence set of Ferrers boards by specifying that two boards are connected by an…
To an arbitrary directed graph we associate a row-finite directed graph whose C*-algebra contains the C*-algebra of the original graph as a full corner. This allows us to generalize results for C*-algebras of row-finite graphs to…
In this manuscript we develop a version of Szemer\'edi's regularity lemma that is suitable for analyzing multicolorings of complete graphs and directed graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M. Szegedy…
We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and…
The Hajnal--Szemer\'edi theorem states that for any integer $r \ge 1$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $\delta(G) \ge (1 - 1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the…
Bandelt and Mulder's structural characterization of Bipartite Distance Hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pending vertices or twins (i.e.,…
In this article, we provide three formulas allowing to compute the minimum amount of initial chips leading to an infinite Chip-firing game. These formulas hold for strongly connected directed loop-free multigraphs and generalize what was…