Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for $k$-regular graphs with $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.     

On a conjecture of Gentner and Rautenbach

Gentner and Rautenbach conjectured that the size of a minimum zero forcing set in a connected graph on $n$ vertices with maximum degree $3$ is at most $\frac{1}{3}n+2$. We disprove this conjecture by constructing a collection of connected graphs $\left\{G_n\right\}$ with maximum degree 3 of arbitrarily large order having zero forcing number at least $\frac{4}{9}\left|V\left(G_n\right)\right|$.     

List 3-dynamic coloring of graphs with small maximum average degree

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $min\{r,{deg}_G\left(v\right)\}$ distinct colors in $N_G\left(v\right)$. The $r$-dynamic chromatic number${\chi }_r^d\left(G\right)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list$r$-dynamic chromatic number of a graph $G$ is denoted by $c{h}_r^d\left(G\right)$.Recently, Loeb et al. (0000) showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. (0000) studied the maximum average condition to have ${\chi }_3^d\left(G\right)\le 4,5$, or $6$. On the other hand, Song et al. (2016) showed that if $G$ is planar with girth at least 6, then ${\chi }_r^d\left(G\right)\le r+5$ for any $r\ge 3$.In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $c{h}_3^d\left(G\right)\le 6$ if $mad\left(G\right)<\frac{18}{7}$, $c{h}_3^d\left(G\right)\le 7$ if $mad\left(G\right)<\frac{14}{5}$, and $c{h}_3^d\left(G\right)\le 8$ if $mad\left(G\right)<3$. All of the bounds are tight.     

Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a mulitigraph $G$, denoted ${\chi }_s^{\prime }\left(G\right)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star$k$-edge-colorable if ${\chi }_s^{\prime }\left(G\right)\le k$. Dvo?ák et al. (2013) proved that every subcubic multigraph is star 7-edge-colorable, and conjectured that every subcubic multigraph should be star 6-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7∕3$ is star list-5-edge-colorable. It is known that a graph with maximum average degree $14∕5$ is not necessarily star 5-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12∕5$ is star 5-edge-colorable.     

List r-hued chromatic number of graphs with bounded maximum average degrees

For integers $k,r>0$, a $(k,r)$-coloring of a graph $G$ is a proper coloring $c$ with at most $k$ colors such that for any vertex $v$ with degree $d\left(v\right)$, there are at least min$\{d\left(v\right),r\}$ different colors present at the neighborhood of $v$. The $r$-hued chromatic number of $G$, ${\chi }_r\left(G\right)$, is the least integer $k$ such that a $(k,r)$-coloring of $G$ exists. The list$r$-hued chromatic number${\chi }_{L,r}\left(G\right)$ of $G$ is similarly defined. Thus if $\Delta \left(G\right)\ge r$, then ${\chi }_{L,r}\left(G\right)\ge {\chi }_r\left(G\right)\ge r+1$. We present examples to show that, for any sufficiently large integer $r$, there exist graphs with maximum average degree less than 3 that cannot be $(r+1,r)$-colored. We prove that, for any fraction $q<\frac{14}{5}$, there exists an integer $R=R\left(q\right)$ such that for each $r\ge R$, every graph $G$ with maximum average degree $q$ is list $(r+1,r)$-colorable. We present examples to show that for some $r$ there exist graphs with maximum average degree less than 4 that cannot be $r$-hued colored with less than $\frac{3r}{2}$ colors. We prove that, for any sufficiently small real number $?>0$, there exists an integer $h=h\left(?\right)$ such that every graph $G$ with maximum average degree $4??$ satisfies ${\chi }_{L,r}\left(G\right)\le r+h\left(?\right)$. These results extend former results in Bonamy et al. (2014).     

Further results on (v,4,1)-perfect difference families

In this paper, it is shown that an ASP$(2r+1,3)$ exists for $2\le r\le 100$ and $r\ne 4,5$. The existence of a $(v,4,1)$-PDF is investigated by taking advantage of the relationship between ASPs and perfect difference families (PDFs). It is proved that a $(12t+1,4,1)$-PDF exists for $t\le 100$ and $t\ne 2,3$. Several recursive constructions for ASPs and PDFs are also presented. As a consequence, the existence results of an optimal $(v,4,1)$-OOC is updated.     

On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators $H(t)=?d^2/d{x}^2+q(x)+t\cos x$ and $H(t)=?d^2/d{x}^2+q(x)+A\cos (tx)$ defined on $L^2(\mathbb{R})$, where q is an even potential that is bounded from below, A is a constant, and $t>0$ is a parameter. We assume that $H(t)$ has at least two eigenvalues below its essential spectrum; and we denote by ${\lambda }_1(t)$ and ${\lambda }_2(t)$ the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap $\Gamma (t)={\lambda }_2(t)?{\lambda }_1(t)$ in the limit as $t\to \infty$.     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).