Acyclic edge coloring

Acyclic edge-coloring using entropy compression

An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected…

A new upper bound on the acyclic chromatic indices of planar graphs

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that $a'(G)\le Δ+2$ for any simple graph $G$ with maximum degree $Δ$. In this paper, we prove that if $G$ is a planar graph, then $a'(G)\leqΔ+7$. This improves a result by Basavaraju et al. [{\em Acyclic edge-coloring of planar graphs}, SIAM J. Discrete Math., 25 (2011), pp. 463-478], which says that every planar graph $G$ satisfies $a…

具有最小和最大可能颜色数的有向图的无环与完全着色

Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors

本文研究有向图无环与完全着色中颜色数的极值问题，构造了任意给定二色数 r 和双色数 t（r≤t）的非对称有向图，并给出非对称情形下的二次上界。

Defective acyclic colorings of planar graphs

This paper studies two variants of defective acyclic coloring of planar graphs. For a graph $G$ and a coloring $φ$ of $G$, a 2CC transversal is a subset $E'$ of $E(G)$ that intersects every 2-colored cycle. Let $k$ be a positive integer. We denote by $m_k(G)$ the minimum integer $m$ such that $G$ has a proper $k$-coloring which has a 2CC transerval of size $m$, and by $m'_k(G)$ the minimum size of a subset $E'$ of $E(G)$ such that $G-E'$ is acyclic $k$-colorable. We prove that for any $n$-vertex $3$-colorable planar graph $G$, $m_3(G) \le n - 3$ and for any planar graph $G$, $m_4(G) \le n - 5$…

On b-acyclic chromatic number of a graph

Let $G$ be a graph. We introduce the acyclic b-chromatic number of $G$ as an analogue to the b-chromatic number of $G$. An acyclic coloring of a graph $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the induced subgraph on vertices of any two colors $i,j\in \{1,\dots,k\}$ induces a forest. On the set of all acyclic colorings of $G$ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number $A(G)$ of $G$ and the maximum cardinality of its minimal el…

The diachromatic number of digraphs

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichroma…

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relati…

Complexity of acyclic colorings of graphs and digraphs with degree and girth constraints

We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…

Acyclic edge coloring of sparse graphs

A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $χ'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. The maximum average degree of a graph $G$, denoted by $\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In this paper, it is proved that if $\mad(G)<4$, then $χ'_a(G)\leq{Δ(G)+2}$; if $\mad(G)<3$, then $χ'_a(G)\leq{Δ(G)+1}$. This implies that every triangle-free planar graph $G$ is acyclically edge $(Δ(G)+2)$-colorable.

Acyclic and Star Colorings of Cographs

An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star ch…

On Acyclic Edge-Coloring of Complete Bipartite Graphs

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $Δ= Δ(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = Δ+ 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus es…

Sectionable Tournaments: Their topology and Coloring

We provide a detailed study of topological and combinatorial properties of sectionable tournaments. This class forms an inductively constructed family of tournaments grounded over simply disconnected tournaments, those tournaments whose fundamental groups of acyclic complexes are non-trivial. When $T$ is a sectionable tournament, we fully describe the cell-structure of its acyclic complex $Acy(T)$ by using the adapted machinery of discrete Morse theory for acyclic complexes of tournaments. In the combinatorial side, we demonstrate that the dimension of the complex $Acy(T)$ has a role to play. …

Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{č}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree $Δ$ is acyclically edge $(Δ+ 2)$-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at le…

Coloring of a Digraph

\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The \emph{dichromatic number} $χ_d(D)$ of $D$ is the minimum number of colors needed in a coloring of $D$. Besides obtaining many results and bounds for $χ_d(D)$ analogous to that of chromatic number of a graph, we prove $χ_d(D)=1$ if $D$ is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained …

Further result on acyclic chromatic index of planar graphs

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq Δ(G) + 2$ for any simple graph $G$ with maximum degree $Δ(G)$. In this paper, we prove that every planar graph $G$ admits an acyclic edge coloring with $Δ(G) + 6$ colors.

An Improved Bound of Acyclic Vertex-Coloring

The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $α>2^{-1/3}$ there exists an integer $Δ_α$ such that if the maximum degree $Δ$ of a graph is at least $Δ_α$, then the acyclic chromatic number of the graph is at most $\lceilαΔ^{4/3} \rceil +Δ+ 1$. The previous best bound, due to Gonçalves et al (2020), was $(3/2) Δ^{4/3} + O(Δ)$.

An improved bound on acyclic chromatic index of planar graphs

Proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $χ'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. Basavaraju et al. [Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2) (2011), 463--478] showed that $χ'_a(G)\le Δ(G)+12$ for planar graphs $G$ with maximum degree $Δ(G)$. In this paper, the bound is improved to $Δ(G)+10$.

Acyclic edge coloring of graphs

An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiamčík (1978) conjectured that $\chiup_{a}'(G) \leq Δ(G) + 2$, where $Δ(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $κ$ is {\em $κ$-deletion-minimal} if $\ch…

Acyclic Orientations and the Chromatic Polynomial of Signed Graphs

We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $χ_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|χ_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplan…

New Bounds for the Acyclic Chromatic Index

An edge coloring of a graph $G$ is called an acyclic edge coloring if it is proper and every cycle in $G$ contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of $G$ is called the acyclic chromatic index of $G$ and is denoted by $a'(G)$. Fiamčik and independently Alon, Sudakov, and Zaks conjectured that $a'(G) \leq Δ(G)+2$, where $Δ(G)$ denotes the maximum degree of $G$. The best known general bound is $a'(G)\leq 4(Δ(G)-1)$ due to Esperet and Parreau. We apply a generalization of the Lovász Local Lemma to show that if $G$ contains no…

有向图中无环集和着色问题在度数和环长限制下的研究

Acyclic sets and colorings in digraphs under restrictions on degrees and cycle lengths

本文研究了有向图中无环集和着色问题在度数和环长限制下的性质，证明了在随机有向图中无环集大小的渐进行为，并提出了相关猜想。

Acyclic chromatic index of triangle-free 1-planar graphs

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq Δ(G) + 2$ for any simple graph $G$ with maximum degree $Δ(G)$. A graph is {\em $1$-planar} if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we prove that every triangle-free $1$-planar graph $G$ has an acyclic edge coloring with $Δ(G) + 16$ color…

3稀疏图的无圈边着色

Acyclic Edge Coloring of 3-sparse Graphs

研究3稀疏图的无圈边着色问题，证明Fiamčík猜想并给出更强的界。

Acyclic Edge Coloring through the Lovász Local Lemma

We give a probabilistic analysis of a Moser-type algorithm for the Lovász Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $Δ$ has an acyclic proper edge coloring with at most $\lceil 3.74(Δ-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(Δ-1)$. The main contribution of this work is that it comprises a probabilistic an…

$d$-Regular graphs of acyclic chromatic index at least $d+2$

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. Alon et.al also raised the question whether the complete graphs of even order are the only regular graphs which require $Δ+2$ colors to be acyclically edge colored. In this paper, using a si…

Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. Fiam${\rm \check{c}}$ik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that $a'(G)\le Δ+ 2$ for any simple graph $G$ with maximum degree $Δ$. Basavaraju and Chandran (2009) showed that every graph $G$ with $Δ=4$, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular g…

无环着色参数的改进界

Improved bounds for acyclic coloring parameters

本文研究了图的无环着色参数，证明了无环色指数最多为$2Δ-1$，并在最大度$Δ$足够大时给出无环色数的新上界，通过改进的随机算法在非正常着色基础上逐步实现正常着色以节省颜色。

The Acyclic Chromatic Index is Less than the Double of the Max Degree

The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2Δ-1$ colors are sufficient to produce such a coloring, where $Δ$ is the maximum degree of the graph. In contrast with most extant randomized algorithmic approaches to the chromatic index, where the algorithms presuppose enough colors to guarantee properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-type algorithm produces a not necessarily proper random color…

Structural properties of 1-planar graphs and an application to acyclic edge coloring

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Therefore, two open problems presented by Fabrici and Madaras [The structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are solved. Furthermore, we prove that each 1-planar graph $G$ with …

Acyclic Coloring of Products of Digraphs and of Digraphs with Bounded Treewidth

The dichromatic number of a digraph $G$ is the smallest integer $χ_a(G)$ such that the vertex set of $G$ can be partitioned into $χ_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of di…

Characterization of forbidden subgraphs for bounded star chromatic number

The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbid…

关于三次图的acyclic b-chromatic number

On acyclic b-chromatic number of cubic graphs

本文研究了三次图的acyclic b-chromatic number，定义了基于acyclic recoloring step的最大颜色数，并探讨了其在传递闭包中的等价表述。

退化图的无环色指数上界

Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs

本研究针对退化图的无环色指数上界问题，证明3-退化图的无环色指数不超过Δ+5，推进了Fiamčík猜想的相关研究，并给出k≥4时k-退化图的无环色指数上界。

On the simultaneous edge coloring of graphs

On the simultaneous edge coloring of graphs

有向图的分数多数着色：界与不可近似性

Fractional majority coloring of digraphs: bounds and inapproximability

该研究针对有向图的分数多数着色问题，通过随机循环匹配采样等方法改进了着色数上界，证明了相关判定问题的NP完全性及不可近似性阈值，还得到了随机近似算法。

Restricted extension of sparse partial edge colorings of complete graphs

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

多数区分边着色

Majority distinguishing edge coloring

本文结合多数着色与区分着色概念，证明无悬挂边图可用至多 ⌈√Δ⌉+5 种颜色实现多数区分边着色，并推广到若干图类及对称有向图。

3-facial edge-coloring of plane graphs

An $\ell$-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most $\ell$ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the $\ell$-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It is conjectured that at most $3\ell + 1$ colors suffice for an $\ell$-facial edge-coloring of any plane graph. The conjecture has only been confirmed for $\ell \le 2$, and in this paper, we prove its validity for $\ell = 3$.

l-facial edge colorings of graphs

An l-facial edge coloring of a plane graph is a coloring of the edges such that any two edges at distance at most l on a boundary walk of some face receive distinct colors. It is conjectured that 3l + 1 colors suffice for an l-facial edge coloring of any plane graph. We prove that 7 colors suffice for a 2-facial edge coloring of any plane graph and therefore confirm the conjecture for l = 2.

Maximal edge colorings of graphs

For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $χ'(K_n)$ colors such that adding any edge to $G$ in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges there are no graphs with a maximal edge coloring, while for some other values, they provided constructions of such graphs. However, for many values of the number of edges determining whether there exists any graph with a maximal edge coloring remained open. We give a complete solution of this problem.

Majority Edge-Colorings of Graphs

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the best possible results that every graph of minimum degree at least $2$ has a majority $4$-edge-coloring, and that every graph of minimum degree at least $4$ has a majority $3$-edge-coloring. Furthermore, we discuss a natural variation of majority edge-colorings and some related open problems.

Vertex-Coloring 2-Edge-Weighting of Graphs

A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in E(G)$. Given a graph $G$ and a vertex coloring $c_0$, does there exist an edge-weighting such that the induced vertex coloring is $c_0$? We investigate this problem by considering edge-weightings defined o…

A conditional greedy algorithm for edge-coloring

We present a novel algorithm for edge-coloring of multigraphs. The correctness of this algorithm for multigraphs with $χ' > Δ+1$ ($χ'$ is the chromatic edge number and $Δ$ is the maximum vertex degree) would prove a long standing conjecture in edge-coloring of multigraphs.

Improved bound on facial parity edge coloring

A facial parity edge coloring of a 2-edge connected plane graph is an edge coloring where no two consecutive edges of a facial walk of any face receive the same color. Additionally, for every face f and every color c either no edge or an odd number of edges incident to f are colored by c. Czap, Jendrol', Kardoš and Sotak showed that every 2-edge connected plane graph admits a facial parity edge coloring with at most 20 colors. We improve this bound to 16 colors.

Linear Hypergraph List Edge Coloring - Generalizations of the EFL Conjecture to List Coloring

Motivated by the Erdős-Faber-Lovász (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 Vizing's 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.

奇度2的三次图的正常6-边着色

Normal 6-edge-colorings of cubic graphs with oddness 2

本文推广了Mazzuoccolo和Mkrtchyan关于循环置换图正常6-边着色的结果，证明了每个奇度为2的三次图都允许用至多6种颜色进行正常边着色。

最大度至多为5的三角剖分的列表边着色

List-Edge-Coloring Triangulations with Maximum Degree at most 5

该研究证明了最大度不超过5的三角剖分满足列表边着色猜想，推进了图论中列表边着色相关问题的研究进展。

Avoiding and extending partial edge colorings of hypercubes

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $φ$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge coloring of $Q_d$ that agrees with the coloring $φ$ on every edge that is colored under $φ$; or, similarly, if there is a proper $d$-edge coloring that disagrees with $φ$ on every edge that is colored under $φ$. In particular, we prove that for any $d\geq 1$, if $φ$ is a partial $d$-edge coloring of $Q_d$, then $φ$ is avoidable if every color appears on at most $d/8$ edg…

Appendix for almost-rainbow edge-colorings of some small subgraphs

This appendix for our article, "Almost-rainbow edge-colorings of some small subgraphs", contains the full proof of Theorem 4.1.

Restricted extension of sparse partial edge colorings of hypercubes

We consider the following type of question: Given a partial proper $d$-edge coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors for the non-colored edges of $Q_d$,can we extend the partial coloring to a proper $d$-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.

Star edge coloring of some classes of graphs

\textit{A star edge coloring} of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.

On edge-colorings of bicubic planar graphs

In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two edge-3-colorings of a planar bicubic graph are Kempe equivalent. Additionaly we show that the degree of colorings behaves well with respect to Kempe equivalence.

Partially normal 5-edge-colorings of cubic graphs

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter $μ_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $|E(G)|-μ_3(G)$, …

Note on list star edge-coloring of subcubic graphs

{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvořák et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).