List Edge Coloring Research Articles

List edge coloring of planar graphs without 6-cycles with two chords

List edge coloring of planar graphs without 6-cycles with two chords

List edge and list total coloring of planar graphs without intersecting 8-cycles

In this thesis, given a planar graph G containing no intersecting 8-cycles, we obtain χ l­'(G)= k and χ"l(G) = k + 1 for k = max{10, Δ(G)}.

List Edge Colorings of Planar Graphs with 7-cycles Containing at Most Two Chords

In this paper we prove that if G is a planar graph, and each 7-cycle contains at most two chords, then G is edge-k-choosable, where k = max{8, ?(G) + 1}.

A Structure of 1-Planar Graph and Its Applications to Coloring Problems

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the $(p,1)$-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the $(p,1)$-total labelling number of every 1-planar graph $G$ is at most $\Delta(G)+2p-2$ provided that $\Delta(G)\geq 8p+2$ and $p\geq 2$, and show that every 1-planar graph has an equitable edge coloring with $k$ colors for any integer $k\geq 18$. These three results respectively generalize the main theorems of three different previously published papers.

Edge‐coloring linear hypergraphs with medium‐sized edges

Motivated by the Erdos̋‐Faber‐Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper‐edge sizes are bounded between i and inclusive, then there is a list edge coloring using colors. The dependence on n in the upper bound is optimal (up to the value of Ci,ϵ).

An upper bound for the choice number of star edge coloring of graphs

The star chromatic index of a multigraph G, denoted χs′(G), 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 χs′(G)≤k. Dvořák et al. (2013) proved that every subcubic multigraph is star 7-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star 6-edge-colorable. In this paper, we consider this problem in a more general setting, we investigate star list edge coloring of general graph G and obtain an upper bound for the choice number of star edge coloring of graphs, namely, we proved that χsl′≤⌈2Δ32(1Δ+2)12+2Δ⌉.

Computing the list chromatic index of graphs

As starting point, we formulate a corollary to the Quantitative Combinatorial Nullstellensatz. This corollary does not require the consideration of any coefficients of polynomials, only evaluations of polynomial functions. In certain situations, our corollary is more directly applicable and more ready-to-go than the Combinatorial Nullstellensatz itself. It is also of interest from a numerical point of view. We use it to explain a well-known connection between the sign of 1-factorizations (edge colorings) and the List Edge Coloring Conjecture. For efficient calculations and a better understanding of the sign, we then introduce and characterize the sign of single 1-factors. We show that the product over all signs of all the 1-factors in a 1-factorization is the sign of that 1-factorization. Using this result in an algorithm, we attempt to prove the List Edge Coloring Conjecture for all graphs with up to 10 vertices. This leaves us with some exceptional cases that need to be attacked with other methods.

The Complexity of (List) Edge-Coloring Reconfiguration Problem

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 computational hardness was unknown for the non-list variant in which every edge has the same list of k colors. In this paper, we first improve the known result by proving that, for every integer \(k \ge 4\), the problem remains PSPACE-complete even for planar graphs of maximum degree three and bounded bandwidth. Since the problem is known to be solvable in polynomial time if \(k \le 3\), our result gives a sharp analysis of the complexity status with respect to the number k of colors. We then give the first computational hardness result for the non-list variant: for every integer \(k \ge 5\), the non-list variant is PSPACE-complete even for planar graphs of maximum degree k and bandwidth linear in k.

The tournament scheduling problem with absences

We study time scheduling problems with allowed absences as a new kind of graph coloring problem. One may think of a sport tournament where each player (each team) is permitted a certain number t of absences. We then examine how many rounds are needed to schedule the whole tournament in the worst case. This upper limit depends on t and on the structure of the graph G whose edges represent the games that have to be played, but also on whether or not the absences are announced before the tournament starts. Therefore, we actually have two upper limits for the number of required rounds. We have χt(G) for pre-scheduling if all absences are pre-fixed, and we have χOLt(G) for on-line scheduling if we have to stay flexible and deal with absences when they occur. We conjecture that χt(G)=Δ(G)+2t and that χOLt(G)=χ′(G)+2t. The first conjecture is stronger than the Total Coloring Conjecture while the second is weaker than the On-Line List Edge Coloring Conjecture. Our conjectures hold for all bipartite graphs. For complete graphs, we prove them partially. Lower and upper bounds to χt(G) and χOLt(G) for general multigraphs G are established, too.

List edge and list total coloring of planar graphs with maximum degree 8

Let $$G$$G be a planar graph with maximum degree $$\varDelta \ge 8$$Δź8 and without chordal 5-cycles. Then $$\chi '_{l}(G)=\varDelta $$źlź(G)=Δ and $$\chi ''_{l}(G)=\varDelta +1$$źlźź(G)=Δ+1.

A note on the minimum number of choosability of planar graphs

The problem of minimum number of choosability of graphs was first introduced by Vizing. It appears in some practical problems when concerning frequency assignment. In this paper, we study two important list coloring, list edge coloring and list total coloring. We prove that $$\chi '_{l}(G)=\varDelta $$?l?(G)=Δ and $$\chi ''_{l}(G)=\varDelta +1$$?l??(G)=Δ+1 for planar graphs with $$\varDelta \ge 8$$Δ?8 and without adjacent 4-cycles.

Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree

We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.

On strong list edge coloring of subcubic graphs

In this paper, we give some new results on strong list edge coloring of subcubic graphs. We prove that every subcubic graph with maximum average degree less than 52, 83 and 145 can be strongly list edge colored with seven, eight and nine colors respectively.

List Edge Coloring of Planar Graphs Without Non-Induced 6-Cycles

A graph $$G$$G is edge-$$L$$L-colorable if for a given edge assignment $$L=\{L(e):e\in E(G)\}$$L={L(e):e?E(G)}, there exits a proper edge-coloring $$\phi $$? of $$G$$G such that $$\phi (e)\in L(e)$$?(e)?L(e) for all $$e\in E(G)$$e?E(G). If $$G$$G is edge-$$L$$L-colorable for every edge assignment $$L$$L with $$| L(e)|\ge k$$|L(e)|?k for $$e\in E(G)$$e?E(G), then $$G$$G is said to be edge-$$k$$k-choosable. In this paper, we prove that if $$G$$G is a planar graph without non-induced $$6$$6-cycles, then $$G$$G is edge-$$k$$k-choosable, where $$k=\max \{8, \Delta (G)+1\}$$k=max{8,Δ(G)+1}.

A note on list edge and list total coloring of planar graphs without adjacent short cycles

Let G be a planar graph with maximum degree Δ. In this paper, we prove that if any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4} in G, then the list edge chromatic number xl′ (G) = Δ and the list total chromatic number xl″ (G) = Δ+1.

List edge and list total colorings of planar graphs without non-induced 7-cycles

Graphs and Algorithms Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.

Strong List Edge Coloring of Subcubic Graphs

We study strong list edge coloring of subcubic graphs, and we prove that every subcubic graph with maximum average degree less than 15/7, 27/11, 13/5, and 36/13 can be strongly list edge colored with six, seven, eight, and nine colors, respectively.

Acyclic list edge coloring of outerplanar graphs

An acyclic list edge coloring of a graph G is a proper list edge coloring such that no bichromatic cycles are produced. In this paper, we prove that an outerplanar graph G with maximum degree Δ≥5 has the acyclic list edge chromatic number equal to Δ.

Reconfiguration of list edge-colorings in a graph

We study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing only one edge color assignment at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. First we show that this problem is PSPACE-complete, even for planar graphs of maximum degree 3 and just six colors. We then consider the problem restricted to trees. We show that any list edge-coloring can be transformed into any other under the sufficient condition that the number of allowed colors for each edge is strictly larger than the degrees of both its endpoints. This sufficient condition is best possible in some sense. Our proof yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices using O(n2) recolor steps. This worst-case bound is tight: we give an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n2) recolor steps.

An Improved Sufficient Condition for Reconfiguration of List Edge-Colorings in a Tree

We study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing only one edge color assignment at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. Ito, Kaminski and Demaine gave a sufficient condition so that any list edge-coloring of a tree can be transformed into any other. In this paper, we give a new sufficient condition which improves the known one. Our sufficient condition is best possible in some sense. The proof is constructive, and yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices via O(n2) recoloring steps. We remark that the upper bound O(n2) on the number of recoloring steps is tight, because there is an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n2) recoloring steps.