arXiv subjects

André Raspaud

Publications and source records attributed to André Raspaud.

11 recordsLinked to original sources

arXiv2021

Star coloring of sparse graphs

A proper coloring of the vertices of a graph is called a \emph{star coloring} if the union of every two color classes induces a star forest. The star chromatic number $χ_s(G)$ is the smallest number of colors required to obtain a star coloring of $G$. In this paper, we study the relationship between the star chromatic number $χ_s(G)$ and the maximum average degree $\mbox{Mad}(G)$ of a graph $G$. We prove that: (1) If $G$ is a graph with $\mbox{Mad}(G) < \frac{26}{11}$, then $χ_s(G)\leq 4$. (2) If $G$ is a graph with $\mbox{Mad}(G) < \frac{18}{7}$ and girth at least 6, then $χ_s(G)\leq 5$. (3) If $G$ is a graph with $\mbox{Mad}(G) < \frac{8}{3}$ and girth at least 6, then $χ_s(G)\leq 6$. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets.

math.CO↗

arXiv2021

Injective edge-coloring of graphs with given maximum degree

A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index of $G$, $χ'_{inj}(G)$, is the minimum number of colors needed for an injective edge-coloring of $G$. We study how large can be the injective chromatic index of $G$ in terms of maximum degree of $G$ when we have restrictions on girth and/or chromatic number of $G$. We also compare our bounds with analogous bounds on the strong chromatic index.

math.CO↗

arXiv2020

A note on adaptable choosability and choosability with separation of planar graphs

Let $F$ be a (possibly improper) edge-coloring of a graph $G$; a vertex coloring of $G$ is \emph{adapted to} $F$ if no color appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph $G$ is such that given any list assignment $L$ to the vertices of $G$, with $|L(v)| \ge k$ for all $v$, and any edge-coloring $F$ of $G$, $G$ admits a coloring $c$ adapted to $F$ where $c(v) \in L(v)$ for all $v$, then $G$ is said to be \emph{adaptably $k$-choosable}. A {\em $(k,d)$-list assignment} for a graph $G$ is a map that assigns to each vertex $v$ a list $L(v)$ of at least $k$ colors such that $|L(x) \cap L(y)| \leq d$ whenever $x$ and $y$ are adjacent. A graph is {\em $(k,d)$-choosable} if for every $(k,d)$-list assignment $L$ there is an $L$-coloring of $G$. It has been conjectured that planar graphs are $(3,1)$-choosable. We give some progress on this conjecture by giving sufficient conditions for a planar graph to be adaptably $3$-choosable. Since $(k,1)$-choosability is a special case of adaptable $k$-choosablity, this implies that a planar graph satisfying these conditions is $(3,1)$-choosable.

math.CO↗

arXiv2020

On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $χ'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if $b=2k+1$ then $χ'_{st}(G) \leq 3k+2$; and if $b=2k$, then $χ'_{st}(G) \leq 3k$; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most $6$; in particular we settle this conjecture for cubic Halin graphs.

math.CO↗

arXiv2018

Cut-edges and regular factors in regular graphs of odd degree

We study $2k$-factors in $(2r+1)$-regular graphs. Hanson, Loten, and Toft proved that every $(2r+1)$-regular graph with at most $2r$ cut-edges has a $2$-factor. We generalize their result by proving for $k\le(2r+1)/3$ that every $(2r+1)$-regular graph with at most $2r-3(k-1)$ cut-edges has a $2k$-factor. Both the restriction on $k$ and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly $2r-3(k-1)+1$ cut-edges but no $2k$-factor. For $k>(2r+1)/3$, there are graphs without cut-edges that have no $2k$-factor, as studied by Bollobás, Saito, and Wormald.

math.CO↗

arXiv2018

Strong edge-colorings of sparse graphs with large maximum degree

A {\em strong $k$-edge-coloring} of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The {\em strong chromatic index} $χ_s'(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ admits a strong $k$-edge-coloring. We give bounds on $χ_s'(G)$ in terms of the maximum degree $Δ(G)$ of a graph $G$. when $G$ is sparse, namely, when $G$ is $2$-degenerate or when the maximum average degree ${\rm Mad}(G)$ is small. We prove that the strong chromatic index of each $2$-degenerate graph $G$ is at most $5Δ(G) +1$. Furthermore, we show that for a graph $G$, if ${\rm Mad}(G)< 8/3$ and $Δ(G)\geq 9$, then $χ_s'(G)\leq 3Δ(G) -3$ (the bound $3Δ(G) -3$ is sharp) and if ${\rm Mad}(G)<3$ and $Δ(G)\geq 7$, then $χ_s'(G)\leq 3Δ(G)$ (the restriction ${\rm Mad}(G)<3$ is sharp).

math.CO↗

arXiv2016

The chromatic number of a signed graph

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $ϕ\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $ϕ(u)$ is different from the colour $σ(uv)ϕ(v)$, where is $σ(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks theorem to signed graphs.

math.CO↗

arXiv2015

Incidence coloring of graphs with high maximum average degree

An incidence of an undirected graph G is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge of $G$ incident with $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent if one of the following holds: (i) $v = w$, (ii) $e = f$ or (iii) $vw = e$ or $f$. An incidence coloring of $G$ assigns a color to each incidence of $G$ in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama \emph{et al.}~\citep{ds05} proved that every graph with maximum average degree strictly less than $3$ can be incidence colored with $Δ+3$ colors. Recently, Bonamy \emph{et al.}~\citep{Bonamy} proved that every graph with maximum degree at least $4$ and with maximum average degree strictly less than $\frac{7}{3}$ admits an incidence $(Δ+1)$-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between $4$ and $6$. In particular we prove that every graph with maximum degree at least $7$ and with maximum average degree less than $4$ admits an incidence $(Δ+3)$-coloring. This result implies that every triangle-free planar graph with maximum degree at least $7$ is incidence $(Δ+3)$-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence $(Δ+ 7)$-coloring. More generally, we prove that $Δ+k-1$ colors are enough when the maximum average degree is less than $k$ and the maximum degree is sufficiently large.

cs.DM↗

arXiv2012

On the size of identifying codes in triangle-free graphs

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let $\M(G)$ be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $Δ\geq 3$, $\M(G)\le n-\tfrac{n}{Δ+o(Δ)}$. This bound is asymptotically tight up to constants due to various classes of graphs including $(Δ-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{Δ-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}Δ+c$ holds for any nontrivial connected identifiable graph $G$.

cs.DM↗

arXiv2006

Vertex labeling and routing in expanded Apollonian networks

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their vertices that allows to determine a shortest path routing between any two vertices of the network based only on the labels.

physics.soc-ph↗

arXiv2004

Recursive graphs with small-world scale-free properties

We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution. We determine relevant characteristics of those graphs: the diameter, degree distribution, and clustering parameter. The graphs have also an interesting recursive property, and generalize recent constructions with fixed degree distributions.

cond-mat.stat-mech↗