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Register a new userLinear Hypergraph List Edge Coloring - Generalizations of the EFL Conjecture to List Coloring
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Vance Faber
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Motivated by the ErdH{o}s-Faber-Lovasz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizings theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the list edge chromatic number is at most 2 times the maximum degree plus 1. We show that for sufficiently large fixed rank and sufficiently large degree, the conjectures are true.
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Motivated by the Erdos-Faber Lovasz conjecture (EFL) for hypergraphs, we explore relationships between several conjectures on the edge coloring of linear hypergraphs. In particular, we are able to increase the class of hypergraphs for which EFL is true.
A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $chi_{s}(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. ErdH{o}s and Nev{s}etv{r}il conjectured in 1985 that the upper bound of $chi_{s}(G)$ is $frac{5}{4}Delta^{2}$ when $Delta$ is even and $frac{1}{4}(5Delta^{2}-2Delta +1)$ when $Delta$ is odd, where $Delta$ is the maximum degree of $G$. The conjecture is proved right when $Deltaleq3$. The best known upper bound for $Delta=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $Delta=4$ the upper bound of list strong chromatic index is 22.
Let $G$ be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of $k$ colors. Suppose that we are given two list edge-colorings $f_0$ and $f_r$ of $G$, and asked whether there exists a sequence of list edge-colorings of $G$ between $f_0$ and $f_r$ such that each list edge-coloring can be obtained from the previous one by changing a color assignment of exactly one edge. This problem is known to be PSPACE-complete for every integer $k ge 6$ and planar graphs of maximum degree three, but any complexity hardness was unknown for the non-list variant. In this paper, we first improve the known result by proving that, for every integer $k ge 4$, the problem remains PSPACE-complete even if an input graph is planar, bounded bandwidth, and of maximum degree three. We then give the first complexity hardness result for the non-list variant: for every integer $k ge 5$, we prove that the non-list variant is PSPACE-complete even if an input graph is planar, of bandwidth linear in $k$, and of maximum degree $k$.
An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $min{r, deg_G(v) }$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $chi_r^d(G)$ 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 $ch_r^d(G)$. Loeb et al. $[11]$ showed that $ch_3^d(G)leq 10$ for every planar graph $G$, and there is a planar graph $G$ with $chi_3^d(G)= 7$. In this paper, we study a special class of planar graphs which have better upper bounds of $ch_3^d(G)$. We prove that $ch_3^d(G) leq 6$ if $G$ is a planar graph which is near-triangulation, where a near-triangulation is a planar graph whose bounded faces are all 3-cycles.
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $vin V(G)$, decide whether $G$ has a proper list coloring; (2) Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $lambda_P: X rightarrow Q$ for $X subseteq V(G),$ decide whether $lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q.$ For Problem 1 we design an $O^*(2^k)$-time randomized algorithm and for Problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. (IWOCA 2019) proved the the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v in V(G),$ decide whether there is a proper list coloring for $G.$ We obtain a kernel with $O(k^2)$ vertices and colors and a compression to a variation of the problem with $O(k)$ vertices and $O(k^2)$ colors.
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