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. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).     

Edge-partitions of graphs and their neighbor-distinguishing index

A proper edge coloring is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The minimum number of colors needed for a neighbor-distinguishing edge coloring is the neighbor-distinguishing index, denoted by ${\chi }_a^{\prime }\left(G\right)$. A graph is normal if it contains no isolated edges. Let $G$ be a normal graph, and let $\Delta \left(G\right)$ and ${\chi }^{\prime }\left(G\right)$ denote the maximum degree and the chromatic index of $G$, respectively. We modify the previously known techniques of edge-partitioning to prove that ${\chi }_a^{\prime }\left(G\right)\le 2{\chi }^{\prime }\left(G\right)$, which implies that ${\chi }_a^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$. This improves the result in Wang et al. (2015), which states that ${\chi }_a^{\prime }\left(G\right)\le \frac{5}{2}\Delta \left(G\right)$ for any normal graph. We also prove that ${\chi }_a^{\prime }\left(G\right)\le 2\Delta \left(G\right)$ when $\Delta \left(G\right)=2^k$, $k$ is an integer with $k\ge 2$.     

Multi-set neighbor distinguishing 3-edge coloring

Let $G$ be a graph without isolated edges, and let $c:E\left(G\right)\to \{1,\dots ,k\}$ be a coloring of the edges, where adjacent edges may be colored the same. The color code of a vertex $v$ is the ordered $k$-tuple $(a_1,a_2,\dots ,a_k)$, where $a_i$ is the number of edges incident with $v$ that are colored $i$. If every two adjacent vertices of $G$ have different color codes, such a coloring is called multi-set neighbor distinguishing. In this paper, we prove that three colors are sufficient to produce a multi-set neighbor distinguishing edge coloring for every graph without isolated edges.     

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.     

An improved upper bound on the adjacent vertex distinguishing total chromatic number of graphs

An adjacent vertex distinguishing total $k$-coloring of a graph $G$ is a proper total $k$-coloring of $G$ such that any pair of adjacent vertices have different sets of colors. The minimum number $k$ needed for such a total coloring of $G$ is denoted by ${\chi }_a^{\prime \prime }\left(G\right)$. In this paper we prove that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)?1$ if $\Delta \left(G\right)\ge 4$, and ${\chi }_a^{\prime \prime }\left(G\right)\le ?\frac{5\Delta \left(G\right)+8}{3}?$ in general. This improves a result in Huang et al. (2012) which states that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)$ for any graph with $\Delta \left(G\right)\ge 3$.     

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).     

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define $V_i?\{v\in V\left(G\right):c\left(v\right)=i\}$ for $i=1$ and $2.$ We say that $G$ is $(d_1,d_2)$-colorable if $G$ has a 2-coloring such that $V_i$ is an empty set or the induced subgraph $G\left[V_i\right]$ has the maximum degree at most $d_i$ for $i=1$ and $2.$ Let $G$ be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for every positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for every positive integer $k.$ In contrast, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4),(3,5),$ and $(2,9).$     

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.     

3-dynamic coloring of planar triangulations

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that any vertex $v$ has 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 $G$.Loeb et al. (2018) showed that if $G$ is a planar graph, then ${\chi }_3^d\left(G\right)\le 10$, and there is a planar graph $G$ with ${\chi }_3^d\left(G\right)=7$. Thus, finding an optimal upper bound on ${\chi }_3^d\left(G\right)$ for a planar graph $G$ is a natural interesting problem. In this paper, we show that ${\chi }_3^d\left(G\right)\le 5$ if $G$ is a planar triangulation. The upper bound is sharp.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

Sum choice number of generalized θ-graphs

Let $G=\left(VE\right)$ be a simple graph and for every vertex $v\in V$ let $L\left(v\right)$ be a set (list) of available colors. $G$ is called $L$-colorable if there is a proper coloring $\phi$ of the vertices with $\phi \left(v\right)\in L\left(v\right)$ for all $v\in V$. A function $f:V\to \mathbb{N}$ is called a choice function of $G$ and $G$ is said to be $f$-list colorable if $G$ is $L$-colorable for every list assignment $L$ choice function is defined by $size\left(f\right)={\sum }_{v\in V}f\left(v\right)$ and the sum choice number ${\chi }_{sc}\left(G\right)$ denotes the minimum size of a choice function of $G$.Sum list colorings were introduced by Isaak in 2002 and got a lot of attention since then.For $r\ge 3$ a generalized ${\theta }_{k_1{k}_2\dots k_r}$-graph is a simple graph consisting of two vertices $v_1$ and $v_2$ connected by $r$ internally vertex disjoint paths of lengths $k_1,k_2,\dots ,k_r$ $(k_1\le k_2\le ?\le k_r)$.In 2014, Carraher et al. determined the sum-paintability of all generalized $\theta$-graphs which is an online-version of the sum choice number and consequently an upper bound for it.In this paper we obtain sharp upper bounds for the sum choice number of all generalized $\theta$-graphs with $k_1\ge 2$ and characterize all generalized $\theta$-graphs $G$ which attain the trivial upper bound $\left|V\left(G\right)\right|+\left|E\left(G\right)\right|$.     

Essential edge connectivity of line graphs

Let $G$ be a graph and $L\left(G\right)$ be its line graph. In 1969, Chartrand and Stewart proved that ${\kappa }^{\prime }\left(L\left(G\right)\right)\ge 2{\kappa }^{\prime }\left(G\right)?2$, where ${\kappa }^{\prime }\left(G\right)$ and ${\kappa }^{\prime }\left(L\left(G\right)\right)$ denote the edge connectivity of $G$ and $L\left(G\right)$ respectively. We show a similar relationship holds for the essential edge connectivity of $G$ and $L\left(G\right)$, written ${\kappa }_e^{\prime }\left(G\right)$ and ${\kappa }_e^{\prime }\left(L\left(G\right)\right)$, respectively. In this note, it is proved that if $L\left(G\right)$ is not a complete graph and $G$ does not have a vertex of degree two, then ${\kappa }_e^{\prime }\left(L\left(G\right)\right)\ge 2{\kappa }_e^{\prime }\left(G\right)?2$. An immediate corollary is that $\kappa \left(L^2\left(G\right)\right)\ge 2\kappa \left(L\left(G\right)\right)?2$ for such graphs $G$, where the vertex connectivity of the line graph $L\left(G\right)$ and the second iterated line graph $L^2\left(G\right)$ are written as $\kappa \left(L\left(G\right)\right)$ and $\kappa \left(L^2\left(G\right)\right)$ respectively.     

A note on the DP-chromatic number of complete bipartite graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo?ák and Postle. Several known bounds for the list chromatic number of a graph $G$, ${\chi }_{?}\left(G\right)$, also hold for the DP-chromatic number of $G$, ${\chi }_{DP}\left(G\right)$. On the other hand, there are several properties of the DP-chromatic number that show that it differs with the list chromatic number. In this note we show one such property. It is well known that ${\chi }_{?}\left(K_{k,t}\right)=k+1$ if and only if $t\ge k^k$. We show that ${\chi }_{DP}\left(K_{k,t}\right)=k+1$ if $t\ge 1+(k^k∕k!)(log(k!)+1)$, and we show that ${\chi }_{DP}\left(K_{k,t}\right)<k+1$ if $t<k^k∕k!$.     

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.