On the (3,1)-choosability of planar graphs without adjacent cycles of length 5,6,7

A $(k,d)$-list assignment $L$ of a graph $G$ is a mapping that assigns to each vertex $v$ a list $L\left(v\right)$ of at least $k$ colors satisfying $|L\left(x\right)\cap L\left(y\right)|\le d$ for each edge $xy$. A graph $G$ is $(k,d)$-choosable if there exists an $L$-coloring of $G$ for every $(k,d)$-list assignment $L$. This concept is also known as choosability with separation. In this paper, we prove that any planar graph $G$ is $(3,1)$-choosable if any $i$-cycle is not adjacent to a $j$-cycle, where $5\le i\le 6$ and $5\le j\le 7$.     

Graphs are (1,Δ+1)-choosable

A graph $G$ is $(k,k^{\prime })$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L\left(v\right)$ of $k$ real numbers, and assigns to each edge $e$ a set $L\left(e\right)$ of $k^{\prime }$ real numbers, there is a total weighting $?:V\left(G\right)\cup E\left(G\right)\to R$ such that $?\left(z\right)\in L\left(z\right)$ for $z\in V\cup E$, and ${\sum }_{e\in E\left(u\right)}?\left(e\right)+?\left(u\right)\ne {\sum }_{e\in E\left(v\right)}?\left(e\right)+?\left(v\right)$ for every edge $uv$. This paper proves that if $G$ is a connected graph of maximum degree $\Delta \ge 2$, then $G$ is $(1,\Delta +1)$-choosable.     

On incidence choosability of cubic graphs

An incidence of a graph $G$ is a pair $(u,e)$ where $u$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $u$. Two incidences $(u,e)$ and $(v,f)$ of $G$ are adjacent whenever (i) $u=v$, or (ii) $e=f$, or (iii) $uv=e$ or $uv=f$. An incidence$k$-coloring of $G$ is a mapping from the set of incidences of $G$ to a set of $k$ colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, has attracted a lot of attention by many authors.On a list version of incidence coloring, it was shown by Benmedjdoub et al. (2017) that every Hamiltonian cubic graph is incidence 6-choosable. In this paper, we show that every cubic (loopless) multigraph is incidence 6-choosable. As a direct consequence, it implies that the list strong chromatic index of a $(2,3)$-bipartite graph is at most 6, where a (2,3)-bipartite graph is a bipartite graph such that one partite set has maximum degree at most 2 and the other partite set has maximum degree at most 3.     

List star edge coloring of k-degenerate graphs

A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Deng et al. and Bezegová et al. independently show that the star chromatic index of a tree with maximum degree $\Delta$ is at most $?\frac{3\Delta }{2}?$, which is tight. In this paper, we study the list star edge coloring of $k$-degenerate graphs. Let $c{h}_{st}^{\prime }\left(G\right)$ be the list star chromatic index of $G$: the minimum $s$ such that for every $s$-list assignment $L$ for the edges, $G$ has a star edge coloring from $L$. By introducing a stronger coloring, we show with a very concise proof that the upper bound on the star chromatic index of trees also holds for list star chromatic index of trees, i.e. $c{h}_{st}^{\prime }\left(T\right)\le ?\frac{3\Delta }{2}?$ for any tree $T$ with maximum degree $\Delta$. And then by applying some orientation technique we present two upper bounds for list star chromatic index of $k$-degenerate graphs.     

Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.     

A note on one-sided interval edge colorings of bipartite graphs

For a bipartite graph G with parts X and Y, an X-interval coloring is a proper edge coloring of G by integers such that the colors on the edges incident to any vertex in X form an interval. Denote by ${\chi }_{int}^{\prime }(G,X)$ the minimum k such that G has an X-interval coloring with k colors. Casselgren and Toft (2016) [12] asked whether there is a polynomial $P(\mathrm{\Delta })$ such that if G has maximum degree at most Δ, then ${\chi }_{int}^{\prime }(G,X)\le P(\mathrm{\Delta })$. In this short note, we answer this question in the affirmative; in fact, we prove that a cubic polynomial suffices. We also deduce some improved upper bounds on ${\chi }_{int}^{\prime }(G,X)$ for bipartite graphs with small maximum degree.     

A tight bound for conflict-free coloring in terms of distance to cluster

Given an undirected graph $G=(V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of G, denoted by ${\chi }_{ON}(G)$.In previous work [WG 2020], we showed the upper bound ${\chi }_{ON}(G)\le \mathsf{dc}(G)+3$, where $\mathsf{dc}(G)$ denotes the distance to cluster parameter of G. In this paper, we obtain the improved upper bound of ${\chi }_{ON}(G)\le \mathsf{dc}(G)+1$. We also exhibit a family of graphs for which ${\chi }_{ON}(G)>\mathsf{dc}(G)$, thereby demonstrating that our upper bound is tight.     

A local condition for k-contractible edges

An edge of a $k$-connected graph is said to be $k$-contractible if the contraction of the edge results in a $k$-connected graph. For a graph $G$ and a vertex $x$ of $G$, let $G\left[N_G\left(x\right)\right]$ be the subgraph induced by the neighborhood of $x$. We prove that if $G\left[N_G\left(x\right)\right]$ has less than $?\frac{k}{2}?$ edges for any vertex $x$ of a $k$-connected graph $G$, then $G$ has a $k$-contractible edge. We also show that the bound $?\frac{k}{2}?$ is sharp.     

Relation between the H-rank of a mixed graph and the rank of its underlying graph

Given a simple graph $G=(V_G,E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\stackrel{?}{G}$ is obtained from $G$ by orienting some of its edges. Let $H\left(\stackrel{?}{G}\right)$ denote the Hermitian adjacency matrix of $\stackrel{?}{G}$ and $A\left(G\right)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\stackrel{?}{G}$ (resp. $G$), written as $rk\left(\stackrel{?}{G}\right)$ (resp. $r\left(G\right)$), is the rank of $H\left(\stackrel{?}{G}\right)$ (resp. $A\left(G\right)$). Denote by $d\left(G\right)$ the dimension of cycle space of $G$, that is $d\left(G\right)=\left|E_G\right|?\left|V_G\right|+\omega \left(G\right)$, where $\omega \left(G\right)$ denotes the number of connected components of $G$. In this paper, we concentrate on the relation between the $H$-rank of $\stackrel{?}{G}$ and the rank of $G$. We first show that $?2d\left(G\right)?rk\left(\stackrel{?}{G}\right)?r\left(G\right)?2d\left(G\right)$ for every mixed graph $\stackrel{?}{G}$. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in Luo et al. (2018); Wong et al. (2016) may be deduced consequently.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.     

Monochromatic k-edge-connection colorings of graphs

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\ge 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is said to be uniformly monochromatic $k$-edge-connected if every two distinct vertices are connected by at least $k$ edge-disjoint monochromatic paths such that all edges of these $k$ paths are colored with a same color. We use $m{c}_k\left(G\right)$ and $um{c}_k\left(G\right)$ to denote the maximum number of colors that ensures $G$ to be monochromatic $k$-edge-connected and, respectively, $G$ to be uniformly monochromatic $k$-edge-connected. In this paper, we first conjecture that for any $k$-edge-connected graph $G$, $m{c}_k\left(G\right)=e\left(G\right)-e\left(H\right)+\lfloor \frac{k}{2}\rfloor$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$. We verify the conjecture for $k=2$. We also prove the conjecture for $G=K_{k+1}$ and $G=K_{k,n}$ with $n\ge k\ge 3$. When $G$ is a minimal $k$-edge-connected graph, we give an upper bound of $m{c}_k\left(G\right)$, i.e., $m{c}_k\left(G\right)\le k-1$. For the uniformly monochromatic $k$-edge-connectivity, we prove that for all $k$, $um{c}_k\left(G\right)=e\left(G\right)-e\left(H\right)+1$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$.     

Remarks on the distribution of colors in Gallai colorings

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \cdots \ge e_k$ of positive integers is an $(n,k)$-sequence if ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i\le k$. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer $k\ge 3$ there exists an integer $g\left(k\right)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g\left(k\right)$. They showed that $g\left(3\right)=5,g\left(4\right)=8$ and $2k-2\le g\left(k\right)\le 8k^2+1$.We show that $g\left(5\right)=10$ and give almost matching lower and upper bounds for $g\left(k\right)$ by showing that with suitable constants $\alpha ,\beta >0$, $\frac{\alpha k^{1.5}}{lnk}\le g\left(k\right)\le \beta k^{1.5}$ for all sufficiently large $k$.     

Connected k-dominating graphs

For a graph $G=(V,E)$, the $k$-dominating graph $D_k\left(G\right)$ of $G$ has vertices corresponding to the dominating sets of $G$ having cardinality at most $k$, where two vertices of $D_k\left(G\right)$ are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote the domination and upper domination numbers of $G$ by $\gamma \left(G\right)$ and $\Gamma \left(G\right)$, respectively, and the smallest integer $\epsilon$ for which $D_k\left(G\right)$ is connected for all $k\ge \epsilon$ by $d_0\left(G\right)$. It is known that $\Gamma \left(G\right)+1\le d_0\left(G\right)\le \left|V\right|$, but constructing a graph $G$ such that $d_0\left(G\right)>\Gamma \left(G\right)+1$ appears to be difficult.We present two related constructions. The first construction shows that for each integer $k\ge 3$ and each integer $r$ such that $1\le r\le k?1$, there exists a graph $G_{k,r}$ such that $\Gamma \left(G_{k,r}\right)=k$, $\gamma \left(G_{k,r}\right)=r+1$ and $d_0\left(G_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)?1$. The second construction shows that for each integer $k\ge 3$ and each integer $r$ such that $1\le r\le k?1$, there exists a graph $Q_{k,r}$ such that $\Gamma \left(Q_{k,r}\right)=k$, $\gamma \left(Q_{k,r}\right)=r$ and $d_0\left(Q_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)$.     

On coloring numbers of graph powers

The weak $r$-coloring numbers ${\mathrm{wcol}}_r\left(G\right)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $\mathrm{col}\left(G\right)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let $G^p$ denote the $p$th power of $G$. We show that, all integers $p>0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta \left(G\right)\le \Delta$ satisfy $\mathrm{col}\left(G^p\right)\in O\left(p\cdot {\mathrm{wcol}}_{\lceil p∕2\rceil }\left(G\right){(\Delta -1)}^{\lfloor p∕2\rfloor }\right)$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2<\mathrm{mad}\left(G\right)\le 2k$, then $\mathrm{col}\left(G^2\right)\le (2k-1)\Delta \left(G\right)+2k+1$.