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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 $\chi_s(G)$ 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 $\chi_s(G)$ 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 $\chi_s(G)\leq 4$. (2) If $G$ is a graph with $\mbox{Mad}(G) < \frac{18}{7}$ and girth at least 6, then $\chi_s(G)\leq 5$. (3) If $G$ is a graph with $\mbox{Mad}(G) < \frac{8}{3}$ and girth at least 6, then $\chi_s(G)\leq 6$. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets., Comment: 20 pages, 5 figures

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., Comment: To appear in JCMCC

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 $G$, $\chi'_{inj}(G)$, 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., Comment: 14 pages

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 $\chi'_{st}(G)$ 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 $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; 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., Comment: 18 pages

An injective edge-coloring $c$ of a graph $G$ is an edge-coloring such that if $e_1$, $e_2$, and $e_3$ are three consecutive edges in $G$ (they are consecutive if they form a path or a cycle of length three), then $e_1$ and $e_3$ receive different colors. The minimum integer $k$ such that, $G$ has an injective edge-coloring with $k$ colors, is called the injective chromatic index of $G$ ($\chi'_{\textrm{inj}}(G)$). This parameter was introduced by Cardoso et \textit{al.} \cite{CCCD} motivated by the Packet Radio Network problem. They proved that computing $\chi'_{\textrm{inj}}(G)$ of a graph $G$ is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. The obtained general bound gives 8 for the injective chromatic index of a subcubic graph. If the graph is subcubic bipartite we improve this last bound. We prove that a subcubic bipartite graph has an injective chromatic index bounded by $6$. We also prove that if $G$ is a subcubic graph with maximum average degree less than $\frac{7}{3} $ (resp. $\frac{8}{3} $, $3$), then $G$ admits an injective edge-coloring with at most 4 (resp. $6$, $7$) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

Using analytical expressions for the pressure and velocity waveforms in tapered vessels, we construct a linear 1D model for wave propagation in stenotic vessels in the frequency domain. We demonstrate that using only two parameters to approximate the exact geometry of the constriction (length and degree of stenosis), we can construct a model that can be solved analytically and can approximate with excellent accuracy the response of the original vessel for a wide range of physiologically relevant frequencies. We then proceed to compare the 1D results with full 3D FSI results from the literature for parameters corresponding to an idealized stenotic carotid artery. We find excellent matching with the volume flow rate over the cardiac cycle (less than $1\%$ error). Using results from DNS simulations to parametrize the velocity profile in the stenotic region, we manage to predict also the pressure distribution with small error (a few percentage points). The method proposed in the paper can be used to approximate vessels of arbitrary shape profile and can be extended to cover the whole cardiovascular tree. Recursive expressions make the solution very fast and open the possibility of carrying out sensitivity and uncertainty quantification studies that require thousands (or even millions) of simulations with minimal cost.

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 Bollob\'as, Saito, and Wormald., Comment: 9 pages

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} $\chi_s'(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ admits a strong $k$-edge-coloring. We give bounds on $\chi_s'(G)$ in terms of the maximum degree $\Delta(G)$ 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 $5\Delta(G) +1$. Furthermore, we show that for a graph $G$, if ${\rm Mad}(G)< 8/3$ and $\Delta(G)\geq 9$, then $\chi_s'(G)\leq 3\Delta(G) -3$ (the bound $3\Delta(G) -3$ is sharp) and if ${\rm Mad}(G)<3$ and $\Delta(G)\geq 7$, then $\chi_s'(G)\leq 3\Delta(G)$ (the restriction ${\rm Mad}(G)<3$ is sharp).

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 $\Delta+3$ 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 $(\Delta+1)$-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 $(\Delta+3)$-coloring. This result implies that every triangle-free planar graph with maximum degree at least $7$ is incidence $(\Delta+3)$-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence $(\Delta + 7)$-coloring. More generally, we prove that $\Delta+k-1$ colors are enough when the maximum average degree is less than $k$ and the maximum degree is sufficiently large.

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ 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., Comment: 9 pages, 0 figures

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $\pi$ that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most $d$ neighbors of $v$ receive color $\pi(v)$. A graph $G$ is said to be $(k,d)^*$-choosable if it admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al. \cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are $(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09} proved that planar graphs without 4- and l-cycles are $(3,1)^*$-choosable, where $l\in \{8,9\}$. There exist planar graphs containing 4-cycles that are not $(3,1)^*$-choosable (Crown, Crown and Woodall, 1986 \cite{CCW-86}). This partly explains the fact that in all above known sufficient conditions for the $(3,1)^*$-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3- and 4-cycles is $(3,1)^*$-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang \cite{XZ-07} and show that every planar graph without 4-cycles is $(3,1)^*$-choosable.

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 $\Delta\geq 3$, $\M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}$. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}{\Delta}+c$ holds for any nontrivial connected identifiable graph $G$.

We prove that for any positive integer $k$, the edges of any graph whose fractional arboricity is at most $k + 1/(3k+2)$ can be decomposed into $k$ forests and a matching.

Sport sciences researchers talk about a relative age effect when they observe a biased distribution of elite athletes' birthdates, with an over-representation of those born at the beginning of the competitive year and an under-representation of those born at the end. Using the whole sample of the French male licensed soccer players (n = 1,831,524), our study suggests that there could be an important bias in the statistical test of this effect. This bias could in turn lead to falsely conclude to a systemic discrimination in the recruitment of professional players. Our findings question the accuracy of past results concerning the existence of this effect at the elite level.

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., Comment: 16 pages, 2 figures