Strong list-chromatic index of subcubic graphs

A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by ${\chi }_s^{\prime }\left(G\right)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erd?s and Ne?et?il conjectured that ${\chi }_s^{\prime }\left(G\right)\le \frac{5}{4}\Delta {\left(G\right)}^2$, where $\Delta \left(G\right)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.     

On stable cutsets in claw-free graphs and planar graphs

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let $K_4$ and $K_{1,3}$ (claw) denote the complete (bipartite) graph on 4 and $1+3$ vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a $K_4$-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, $K_4$)-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs.Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for $K_4$-free planar graphs with maximum degree five.     

Strong edge-coloring for planar graphs with large girth

A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. Let $G$ be a connected planar graph with girth $k\ge 26$ and maximum degree $\Delta$. We show that either $G$ is isomorphic to a subgraph of a very special $\Delta$-regular graph with girth $k$, or $G$ has a strong edge-coloring using at most $2\Delta +\lceil \frac{12(\Delta -2)}{k}\rceil$ colors.     

Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

Let $k$ be a positive integer. An adjacent vertex distinguishing (for short, AVD) total-$k$-coloring of a graph $G$ is a proper total-$k$-coloring of $G$ such that any two adjacent vertices have different color sets, where the color set of a vertex $v$ contains the color of $v$ and the colors of its incident edges. It was conjectured that any graph with maximum degree $\Delta$ has an AVD total-($\Delta +3$)-coloring. The conjecture was confirmed for any graph with maximum degree at most 4 and any planar graph with maximum degree at least 10. In this paper, we verify the conjecture for all planar graphs with maximum degree at least 9. Moreover, we prove that any planar graph with maximum degree at least 10 has an AVD total-($\Delta +2$)-coloring and the bound $\Delta +2$ is sharp.     

List neighbor sum distinguishing edge coloring of subcubic graphs

A proper $k$-edge-coloring of a graph with colors in $\{1,2,\dots ,k\}$ is neighbor sum distinguishing (or, NSD for short) if for any two adjacent vertices, the sums of the colors of the edges incident with each of them are distinct. Flandrin et al. conjectured that every connected graph with at least $6$ vertices has an NSD edge coloring with at most $\Delta +2$ colors. Huo et al. proved that every subcubic graph without isolated edges has an NSD $6$-edge-coloring. In this paper, we first prove a structural result about subcubic graphs by applying the decomposition theorem of Trotignon and Vu?kovi?, and then applying this structural result and the Combinatorial Nullstellensatz, we extend the NSD $6$-edge-coloring result to its list version and show that every subcubic graph without isolated edges has a list NSD $6$-edge-coloring.     

A study on oriented relative clique number

An oriented graph is a directed graph with no directed cycle of length one or two. The relative clique number of an oriented graph is the cardinality of a largest subset $X$ of vertices such that each pair of vertices is either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 32. We also prove an upper bound of 14 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number for the families of outerplanar graphs with girth at least $g$ and planar graphs with girth at least $g+2$ for all $g\ge 3$. Moreover, we study the relation of oriented relative clique number with oriented chromatic number, oriented absolute clique number and maximum degree of a graph. We also show that oriented relative clique number of a connected subcubic graph is at most seven which weakly supports a conjecture by Sopena (JGT 1997).     

Hamiltonicity of edge chromatic critical graphs

Vizing conjectured that every edge chromatic critical graph contains a 2-factor. Believing that stronger properties hold for this class of graphs, Luo and Zhao (2013) showed that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{6n}{7}$ is Hamiltonian. Furthermore, Luo et al. (2016) proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{4n}{5}$ is Hamiltonian. In this paper, we prove that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{3n}{4}$ is Hamiltonian. Our approach is inspired by the recent development of Kierstead path and Tashkinov tree techniques for multigraphs.     

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