Antoine Dailly

Laurent Beaudou, Pierre Coupechoux, Antoine Dailly et al.

Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, an octal game on a path P\_n is equivalent to playing the same octal game on a heap of n counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars.

Nicolas Bousquet, Antoine Dailly, Eric Duchene et al.

We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the problem of finding a coloring with the minimum number of colors where every vertex receives a distinct label. Finding such a coloring generalizes several other well-known problems of vertex-distinguishing colorings in graphs.We show that for any graph (without connected component reduced to an edge or a single vertex), the minimum number of colors for which such a coloring exists can only take 3possible values depending on the order of the graph. Moreover, we provide the exact value for paths, cycles and complete binary trees.

Antoine Dailly, Adriana Hansberg, Denae Ventura

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer $k$ such that every 2-coloring of $K_n$ with more than $k$ edges in each color class contains a balanced copy of $G$, then we say that $G$ is balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who also gave a structural characterization of balanceable graphs. In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.

Antoine Dailly, Florent Foucaud, Adriana Hansberg

A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty-Simon conjecture, states that any diameter-2-critical graph of order n has at most n${}^2$/4 edges, with equality if and only if G is a balanced complete bipartite graph. Many partial results about this conjecture have been obtained, in particular it is known to hold for all sufficiently large graphs, for all triangle-free graphs, and for all graphs with a dominating edge. In this paper, we discuss ways in which this conjecture can be strengthened. Extending previous conjectures in this direction, we conjecture that, when we exclude the class of complete bipartite graphs and one particular graph, the maximum number of edges of a diameter-2-critical graph is at most ((n -- 1)${}^2$/4) + 1. The family of extremal examples is conjectured to consist of certain twin-expansions of the 5-cycle (with the exception of a set of thirteen special small graphs). Our main result is a step towards our conjecture: we show that the Murty-Simon bound is not tight for non-bipartite diameter-2-critical graphs that have a dominating edge, as they have at most (n${}^2$/4) -- 2 edges. Along the way, we give a shorter proof of the Murty-Simon conjecture for this class of graphs, and stronger bounds for more specific cases. We also characterize diameter-2-critical graphs of order n with maximum degree n -- 2: they form an interesting family of graphs with a dominating edge and 2n -- 4 edges.

Antoine Dailly, Julien Moncel, Aline Parreau

The present paper deals with connected subtraction games in graphs, which are generalization of takeaway games. In a connected subtraction game, two players alternate removing a connected sub-graph from a given connected game-graph, provided the resulting graph is connected, and provided the number of vertices of the removed subgraph belongs to a prescribed set of integers. We derive general periodicity results on such games, as well as specific results when played on subdivided stars.

Antoine Dailly, Valentin Gledel, Marc Heinrich

The game Arc-Kayles is played on an undirected graph with two players taking turns deleting an edge and its endpoints from the graph. We study a generalization of this game, Weighted Arc Kayles (WAK for short), played on graphs with counters on the vertices. The two players alternate choosing an edge and removing one counter on both endpoints. An edge can no longer be selected if any of its endpoints has no counter left. The last player to play a move wins. We give a winning strategy for WAK on trees of depth 2. Moreover, we show that the Grundy values of WAK and Arc-Kayles are unbounded. We also prove a periodicity result on the outcome of WAK when the number of counters is fixed for all the vertices but one. Finally, we show links between this game and a variation of the non-attacking queens game on a chessboard.

Antoine Dailly, Eric Duchene, Urban Larsson et al.

We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague-Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague-Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass `break-without-take'.

Antoine Dailly, Florent Foucaud, Anni Hakanen

In the Metric Dimension problem, one asks for a minimum-size set $R$ of vertices such that for any pair of vertices of the graph, there is a vertex from $R$ whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (non-trivially) extends a previous algorithm for oriented trees. We then extend the method to orientations of unicyclic graphs. We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that Metric Dimension is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.

Antoine Dailly, ElÅ1/4bieta Sidorowicz

A $k$-edge-weighting of $G$ is a mapping $ω:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $σ_ω:V(G)\longrightarrow \mathbb{N}$ given by$σ_ω(v)=\sum_{u\in N_G(v)}ω(vu)$ for every $v\in V(G)$. The edge-weighting $ω$ is neighbour sum distinguishing if it yields a proper vertex-colouring $σ_ω$, \emph{i.e.}, $σ_ω(u)\neq σ_ω(v)$ for every edge $uv$ of $G$.We investigate a neighbour sum distinguishing edge-weighting with local constraints, namely, we assume that the set of edges incident to a vertex of large degree is not monochromatic. A graph is nice if it has no components isomorphic to $K_2$. We prove that every nice graph with maximum degree at most~5 admits a neighbour sum distinguishing $(Δ(G)+2)$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights. Furthermore, we prove that every nice graph admits a neighbour sum distinguishing $7$-edge-weighting such that all the vertices of degree at least~6 are incident with at least two edges of different weights. Finally, we show that nice bipartite graphs admit a neighbour sum distinguishing $6$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights.