arXiv subjects

Eric Sopena

Publications and source records attributed to Eric Sopena.

30 records · Page 2Linked to original sources

arXiv2017

Equitable neighbour-sum-distinguishing edge and total colourings

With any (not necessarily proper) edge $k$-colouring $γ:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $σ\_γ$ given by $σ\_γ(v)=\sum\_{e\ni v}γ(e)$.A neighbour-sum-distinguishing edge $k$-colouring is an edge colouring whose associated vertex colouring is proper.The neighbour-sum-distinguishing index of a graph $G$ is then the smallest $k$ for which $G$ admitsa neighbour-sum-distinguishing edge $k$-colouring.These notions naturally extends to total colourings of graphs that assign colours to both vertices and edges.We study in this paper equitable neighbour-sum-distinguishing edge colourings andtotal colourings, that is colourings $γ$ for whichthe number of elements in any two colour classes of $γ$ differ by at most one.We determine the equitable neighbour-sum-distinguishing indexof complete graphs, complete bipartite graphs and forests,and the equitable neighbour-sum-distinguishing total chromatic numberof complete graphs and bipartite graphs.

cs.DM↗

arXiv2016

Packing Coloring of Undirected and Oriented Generalized Theta Graphs

The packing chromatic number $χ$ $ρ$ (G) of an undirected (resp. oriented) graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1,..., V k, in such a way that every two distinct vertices in V i are at distance (resp. directed distance) greater than i in G for every i, 1 $\le$ i $\le$ k. The generalized theta graph $Θ$ {\ell} 1,...,{\ell}p consists in two end-vertices joined by p $\ge$ 2 internally vertex-disjoint paths with respective lengths 1 $\le$ {\ell} 1 $\le$ . . . $\le$ {\ell} p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 $\le$ i $\le$ p, {\ell} i = 3}|, and that both these bounds are tight. We then characterize undirected generalized theta graphs with packing chromatic number k for every k $\ge$ 3. We also prove that the packing chromatic number of any oriented generalized theta graph lies between 2 and 5 and that both these bounds are tight.

cs.DM↗

arXiv2016

2-Distance Colorings of Integer Distance Graphs

A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1,. .. , k} such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number $χ$ 2 (G) of G is then the mallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D = {d 1 ,. .. , d k }, the integer distance graph G = G(D) is the infinite graph defined by V (G) = Z and uv $\in$ E(G) if and only if |v -- u| $\in$ D. We study the 2-distance chromatic number of integer distance graphs for several types of sets D. In each case, we provide exact values or upper bounds on this parameter and characterize those graphs G(D) with $χ$ 2 (G(D)) = Δ(G(D)) + 1.

cs.DM↗

arXiv2015

i-MARK: A New Subtraction Division Game

Given two finite sets of integers $S\subseteq\NNN\setminus\{0\}$ and $D\subseteq\NNN\setminus\{0,1\}$,the impartial combinatorial game $\IMARK(S,D)$ is played on a heap of tokens. From a heap of $n$ tokens, each player can moveeither to a heap of $n-s$ tokens for some $s\in S$, or to a heap of $n/d$ tokensfor some $d\in D$ if $d$ divides $n$.Such games can be considered as an integral variant of \MARK-type games, introduced by Elwyn Berlekamp and Joe Buhlerand studied by Aviezri Fraenkel and Alan Guo, for which it is allowed to move from a heap of $n$ tokensto a heap of $\lfloor n/d\rfloor$ tokens for any $d\in D$.Under normal convention, it is observed that the Sprague-Grundy sequence of the game $\IMARK(S,D)$ is aperiodic for any sets $S$ and $D$.However, we prove that, in many cases, this sequence is almost periodic and that the set of winning positions is periodic.Moreover, in all these cases, the Sprague-Grundy value of a heap of $n$ tokens can be computed in time $O(\log n)$.We also prove that, under misère convention, the outcome sequence of these games is purely periodic.

cs.DM↗

arXiv2015

Ice sliding games

This paper deals with sliding games, which are a variant of the better known pushpush game. On a given structure (grid, torus...), a robot can move in a specific set of directions, and stops when it hits a block or boundary of the structure. The objective is to place the minimum number of blocks such that the robot can visit all the possible positions of the structure. In particular, we give the exact value of this number when playing on a rectangular grid and a torus. Other variants of this game are also considered, by constraining the robot to stop on each case, or by replacing blocks by walls.

math.CO↗

arXiv2015

Packing coloring of some undirected and oriented coronae graphs

The packing chromatic number $\pcn(G)$ of a graph $G$ is the smallest integer $k$ such that its set of vertices $V(G)$ can be partitioned into $k$ disjoint subsets $V\_1$, \ldots, $V\_k$, in such a way that every two distinct vertices in $V\_i$ are at distance greater than $i$ in $G$ for every $i$, $1\le i\le k$. For a given integer $p \ge 1$, the generalized corona $G\odot pK\_1$ of a graph $G$ is the graph obtained from $G$ by adding $p$ degree-one neighbors to every vertex of $G$. In this paper, we determine the packing chromatic number of generalized coronae of paths and cycles. Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of generalized coronae of paths and cycles.

cs.DM↗

arXiv2014

Outerplanar and planar oriented cliques

The clique number of an undirected graph $G$ is the maximum order of a complete subgraph of $G$ and is a well-known lower bound for the chromatic number of $G$. Every proper $k$-coloring of $G$ may be viewed as a homomorphism (an edge-preserving vertex mapping) of $G$ to the complete graph of order $k$. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this paper, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen [S. Sen. Maximum Order of a Planar Oclique Is 15. Proc. IWOCA'2012. {\em Lecture Notes Comput. Sci.} 7643:130--142]. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth $k$ for all $k \ge 4$.

math.CO↗

arXiv2013

Incidence coloring game and arboricity of graphs

An incidence of a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge incident to $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent whenever $v = w$, or $e = f$, or $vw = e$ or $f$. The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980-1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alternately color the incidences of a graph, using a given number of colors, in such a way that adjacent incidences get distinct colors. If the whole graph is colored then Alice wins the game otherwise Bob wins the game. The incidence game chromatic number $i_g(G)$ of a graph $G$ is the minimum number of colors for which Alice has a winning strategy when playing the incidence coloring game on $G$. Andres proved that %$\lceil 3/2 Δ(G)\rceil \le $i_g(G) \le 2Δ(G) + 4k - 2$ for every $k$-degenerate graph $G$. %The arboricity $a(G)$ of a graph $G$ is the minimum number of forests into which its set of edges can be partitioned. %If $G$ is $k$-degenerate, then $a(G) \le k \le 2a(G) - 1$. We show in this paper that $i_g(G) \le \lfloor\frac{3Δ(G) - a(G)}{2}\rfloor + 8a(G) - 2$ for every graph $G$, where $a(G)$ stands for the arboricity of $G$, thus improving the bound given by Andres since $a(G) \le k$ for every $k$-degenerate graph $G$. Since there exists graphs with $i_g(G) \ge \lceil\frac{3Δ(G)}{2}\rceil$, the multiplicative constant of our bound is best possible.

cs.DM↗

arXiv2012

The Incidence Chromatic Number of Toroidal Grids

An incidence in a graph $G$ is a pair $(v,e)$ with $v \in V(G)$ and $e \in E(G)$, such that $v$ and $e$ are incident. Two incidences $(v,e)$ and $(w,f)$ are adjacent if $v=w$, or $e=f$, or the edge $vw$ equals $e$ or $f$. The incidence chromatic number of $G$ is the smallest $k$ for which there exists a mapping from the set of incidences of $G$ to a set of $k$ colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\Box C_n$ equals 5 when $m,n \equiv 0 \pmod 5$ and 6 otherwise.

cs.DM↗

arXiv2010

Coloring the square of the Cartesian product of two cycles

The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ in such a way that distinct vertices with distance at most two in $G$ are joined by an edge. We study the chromatic number of the square of the Cartesian product $C_m\Box C_n$ of two cycles and show that the value of this parameter is at most 7 except when $m=n=3$, in which case the value is 9, and when $m=n=4$ or $m=3$ and $n=5$, in which case the value is 8. Moreover, we conjecture that whenever $G=C_m\Box C_n$, the chromatic number of $G^2$ equals $\lceil mn/α(G^2) \rceil$, where $α(G^2)$ denotes the size of a maximal independent set in $G^2$.

cs.DM↗

arXiv2010

Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

The oriented chromatic number of an oriented graph $\vec G$ is the minimum order of an oriented graph $\vev H$ such that $\vec G$ admits a homomorphism to $\vev H$. The oriented chromatic number of an undirected graph $G$ is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph $G$, defined as the minimum order of an oriented graph $\vev U$ such that every orientation $\vec G$ of $G$ admits a homomorphism to $\vec U$. We give some properties of this parameter, derive some general upper bounds on the ordinary and upper oriented chromatic numbers of Cartesian, strong, direct and lexicographic products of graphs, and consider the particular case of products of paths.

cs.DM↗

arXiv2009

Compound Node-Kayles on Paths

In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound, Node-Kayles has been solved for a long time under normal play, while it is still unsolved under misère play. We thus focus on the ten remaining versions, leaving only one of them unsolved.

cs.DM↗