Proper Edge Coloring Research Articles

Proper edge colorings of planar graphs with rainbow C4 ${C}_{4}$‐s

AbstractWe call a proper edge coloring of a graph a B‐coloring if every 4‐cycle of is colored with four different colors. Let denote the smallest number of colors needed for a B‐coloring of . Motivated by earlier papers on B‐colorings, here we consider for planar and outerplanar graphs in terms of the maximum degree . We prove that for planar graphs, for bipartite planar graphs, and for outerplanar graphs with . We conjecture that, for sufficiently large, for planar , and for outerplanar .

Coloring Groups

We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.

On a class of column‐weight 3 decomposable LDPC codes with the analysis of elementary trapping sets

AbstractA column‐weight LDPC code with the parity‐check matrix is called decomposable if there exists a permutation on the rows of , such that can be decomposed into column‐weight one matrix. In this paper, some variations of edge coloring of graphs are used to construct some column‐weight three decomposable LDPC codes with girths at least six and eight. Applying the presented method on several known classes of bipartite graphs, some classes of column‐weight three decomposable LDPC codes are derived having flexibility in length and rate. Interestingly, the constructed parity‐check matrices based on the proper edge coloring of graphs can be considered as the base matrix of some high rate column‐weight three quasi‐cyclic (QC) LDPC codes with maximum‐achievable girth 20. The paper also leads to a simple characterization of elementary trapping sets of the decomposable codes based on the chromatic index of the corresponding normal graphs. This characterization corresponds to a simple search algorithm finds all possible existing elementary trapping sets in a girth‐6 or girth‐8 column‐weight 3 LDPC code which are layered super set of a short cycle in the Tanner graph of the code. Simulation results indicate that the QC‐LDPC codes with large girths lifted from the constructed base matrices have good performances over AWGN channel.

A note on [formula omitted]-coloring and [formula omitted]-coloring 4-regular graphs

Let ∂H(u) be the set of edges incident with a vertex u in the graph H. We say that a graph G is H-colorable if there exist total functions f:E(G)→E(H) and g:V(G)→V(H) such that f is a proper edge-coloring of G and for each vertex u∈V(G) we have f(∂G(u))=∂H(g(u)). Let X¯ be the graph obtained by adding three parallel edges between two degree one vertices of the graph K1,4. Let Aˆ be the graph obtained by adding two pendant edges to two different vertices of a triangle and then adding two edges between the degree two vertex and the two adjacent degree three vertices. Malnegro and Ozeki (2024) [7] asked whether every 4-regular graph with an even number of vertices and an even cycle decomposition admits an X¯-coloring or an Aˆ-coloring and whether every 2-connected planar 4-regular graph with an even number of vertices admits such a coloring. Additionally, they conjectured that for every 2-edge-connected simple cubic graph G with an even number of edges, the line graph L(G) is X¯-colorable. In this short note, we discuss two algorithms for deciding whether a graph G is H-colorable. We give negative answers to the two questions and disprove the conjecture by finding suitable graphs, as verified by two independent algorithms.

A note on interval colourings of graphs

A graph is said to be interval colourable if it admits a proper edge-colouring using palette N in which the set of colours of edges that are incident to each vertex is an interval. The interval colouring thickness of a graph G is the minimum k such that G can be edge-decomposed into k interval colourable graphs. We show that θ(n), the maximum interval colouring thickness of an n-vertex graph, satisfies θ(n)=Ω(log(n)/loglog(n)) and θ(n)⩽n5/6+o(1), which improves on the trivial lower bound and the upper bound given by the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an n-vertex planar graph uses at most 3n/2−2 colours.

On the Anti-Ramsey Threshold for Non-Balanced Graphs

For graphs $G,H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H $ if for every proper edge-coloring of $G$ there is a rainbow copy of $H$, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that the threshold for $G(n,p) \overset{\mathrm{rb}}{\longrightarrow} H$ is at most $n^{-1/m_2(H)}$. Previous results have matched the lower bound for this anti-Ramsey threshold for cycles and complete graphs with at least 5 vertices. Kohayakawa, Konstadinidis and the last author also presented an infinite family of graphs $H$ for which the anti-Ramsey threshold is asymptotically smaller than $n^{-1/m_2(H)}$. In this paper, we devise a framework that provides a richer family of such graphs.

Further split graphs known to be Class 1 and a characterization of subgraph-overfull split graphs

The chromatic index, χ′(G), is the smallest integer k for which a graph G has a proper edge coloring. The Classification Problem involves determining whether a graph G is Class 1 (χ′(G)=Δ(G)) or Class 2 (χ′(G)=Δ(G)+1). It is known that subgraph-overfull graphs must be Class 2. In this paper we are concerned with the Classification Problem for split graphs G[Q∪S] where Q is a clique and S is an independent set. When Δ(G) is odd, G is known to be Class 1. The original proof presented by Chen et al. (1995) has a minor flaw which we detail in this paper while also clarifying that it does not compromise their result. We prove that their technique can be adapted in a non-trivial way to show that some split graphs with even Δ(G) are also Class 1. We show that to solve the Classification Problem for split graphs it suffices to consider that all vertices in Q have degree Δ(G). Considering the subset X of S of the vertices of degree at most Δ(G)/2, we show that if the neighborhood of X has at least ⌊|Q|/2⌋ vertices, then G is Class 1; in the remaining cases we characterize the subgraph-overfull split graphs.

Proper edge-colorings with a rich neighbor requirement

Under a given edge-coloring of a (multi)graph G, an edge is said to be rich if there is no color repetition among its neighboring edges; e.g., any isolated edge is rich. A rich-neighbor coloring of G is a proper edge-coloring such that every non-isolated edge has at least one rich neighbor. For this weaker variant of strong edge-colorings, we believe that every connected subcubic graph ≠K4 admits a rich-neighbor 5-coloring. In support of this, we show that every subcubic graph admits a rich-neighbor 7-coloring. The paper concludes with few open problems for subcubic graphs concerning the analogous notions of normal-neighbor colorings and poor-neighbor colorings.

“Less” Strong Chromatic Indices and the (7, 4)-Conjecture

A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞𝐴(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring. The second is 𝑞𝐵(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring. These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞𝐴(𝐺), 𝑞𝐵(𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special(7, 4)-conjecture. In order to understand the behavior of 𝑞𝐴(𝐺) and 𝑞𝐵(𝐺), we study these parameters for some graphs.We note that 𝑞𝐴(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞𝐵(𝐺) is studied here for the first time.

Proper Edge Colorings of Cartesian Products with Rainbow C_4-s

Related to the famous (7,4)-problem, in an earlier paper we introduced B-colorings. We call a proper edge coloring of a graph G a B-coloring if every 4-cycle of G is colored with four different colors. Let q_B(G) denote the smallest number of colors needed for a B-coloring of G. Here we look at q_B(G) for Cartesian products of paths and cycles. Our main result is that q_B(G) is equal to the chromatic index of G for grids, i.e. for Cartesian products of paths (apart from a few exceptions). This extends an earlier result for the case when G is the d-dimensional cube. Our main tool is a lemma which gives q_B(Gsquare H)le q_B(G)+q_B(H) if chi (G)le q_B(H), chi (H)le q_B(G), where chi (G) is the chromatic number of G.

Equitable Edge Coloring of Splitting Graph of Some Classes of Wheel Graphs

The coloring of all the edges of a graph G with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of W n , D W n and G n by determining its edge chromatic number.

Equitable Edge Coloring of Splitting Graph of Some Classes of Wheel Graphs

The coloring of all the edges of a graph \(G\) with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of \(W_n\), \(DW_n\) and \(G_n\) by determining its edge chromatic number.

Strong Edge Coloring of K4(t)-Minor Free Graphs

A strong edge coloring of a graph G is a proper coloring of edges in G such that any two edges of distance at most 2 are colored with distinct colors. The strong chromatic index χs′(G) is the smallest integer l such that G admits a strong edge coloring using l colors. A K4(t)-minor free graph is a graph that does not contain K4(t) as a contraction subgraph, where K4(t) is obtained from a K4 by subdividing edges exactly t−4 times. The paper shows that every K4(t)-minor free graph with maximum degree Δ(G) has χs′(G)≤(t−1)Δ(G) for t∈{5,6,7} which generalizes some known results on K4-minor free graphs by Batenburg, Joannis de Verclos, Kang, Pirot in 2022 and Wang, Wang, and Wang in 2018. These upper bounds are sharp.

Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs

In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the α-cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a (d,f)-extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given.

The anti-Ramsey threshold of complete graphs

For graphs G and H, let G→rbH denote the property that, for every proper edge-colouring of G, there is a rainbow H in G. For every graph H, the threshold function pHrb=pHrb(n) of this property in the random graph G(n,p) satisfies pHrb=O(n−1/m(2)(H)), where m(2)(H) denotes the so-called maximum 2-density of H. Completing a result of Nenadov, Person, Škorić, and Steger [J. Combin. Theory Ser. B 124 (2017), 1–38], we prove a matching lower bound for pKkrb for k⩾5. Furthermore, we show that pK4rb=n−7/15≪n−1/m(2)(K4).

AVD edge-colorings of cubic Halin graphs

<abstract><p>The adjacent vertex-distinguishing edge-coloring of a graph $ G $ is a proper edge-coloring of $ G $ such that each pair of adjacent vetices receives a distinct set of colors. The minimum number of colors required in an adjacent vertex-distinguishing edge-coloring of $ G $ is called the adjacent vertex-distinguishing chromatic index. In this paper, we determine the adjacent vertex distinguishing chromatic indices of cubic Halin graphs whose characteristic trees are caterpillars.</p></abstract>

Strict neighbor-distinguishing edge coloring of planar graphs

A proper edge coloring of a graph $G$ is strict neighbor-distinguishing if for any two adjacent vertices $u$ and $v$, the set of colors used on the edges incident to $u$ and the set of colors used on the edges incident to $v$ are not included with each other. The strict neighbor-distinguishing index of $G$ is the minimum number $\chi'_{\rm~snd}(G)$ of colors in a strict neighbor-distinguishing edge coloring of $G$. It is conjectured that every simple graph $G$ without leaves satisfies $\chi'_{\rm~snd}(G)\le~2\Delta$ except a special graph $H_{\Delta}$. The currently best-known upper bound is $\chi'_{\rm~snd}(G)\le~3\Delta-1$. An interesting question is: Which graphs $G$ without leaves satisfy $\chi'_{\rm~snd}(G)\le~\Delta+C$, where $C$ is a constant? This paper answers partially this 问题, i.e., it is proved that if $G$ is a planar graph with a girth of at least five, then $\chi'_{\rm~snd}(G)\le~\Delta+25.$

Neighbor Sum Distinguishing Edge (Total) Coloring of Generalized Corona Product

The coloring theory of graphs is an important part of graph theory research. The key problem of the coloring theory of graphs is to determine the coloring number of each kind of coloring. Traditional coloring concepts mainly include proper vertex coloring, proper edge coloring, proper total coloring, and so on. In recent years, scholars at home and abroad have put forward some new coloring concepts, such as neighbor vertex distinguishing edge (total) coloring, and neighbor sum distinguishing edge (total) coloring, based on traditional coloring concepts and by adding other constraints. Some valuable results have been obtained, which further enrich the theory of graph coloring.For a proper [k]-edge coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k]-edge coloring of G. For a proper [k]-total coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k]-total coloring of G . In this paper, the coloring method and coloring index are determined by the process of induction and deduction and the construction of the dyeing method, and then the rationality of the method is verified by inverse proof and mathematical induction.If G is a simple graph with the order n ≥ 5 , and hn = (Hi ) i∈{1,2,…,n} is a sequence of disjoint simple graphs, where every Hi is a simple graph with the order m ≥ 7 . In this paper, we study the neighbor sum distinguishing edge(total) coloring of the generalized corona product G○hn of G and hn . The results are as follows:(1) If G is a path with order n , hn = (Hi ) i∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have (2) If G is a path with order n , hn = (Hi ) i∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have Due to the late development of neighbor sum distinguishing edge (total) coloring of graphs, the related research results are relatively few. By studying the operation graph of a basic simple graph, we can provide the research basis and reference idea for the corresponding coloring of the general graph class. Therefore, it is of theoretical value to study the neighbor sum distinguishing edge (total) coloring problem of generalized corona products of graphs.

Anti-Ramsey threshold of cycles

For graphs G and H, let G⟶rbH denote the property that for every proper edge colouring of G there is a rainbow copy of H in G. Extending a result of Nenadov et al. (2017), we determine the threshold for G(n,p)⟶rbCℓ for cycles Cℓ of any given length ℓ≥4.

The inclusion-free edge-colorings of (3,Δ)-bipartite graphs

An inclusion-free edge-coloring of a graph G with δ ( G ) ≥ 2 is a proper edge-coloring in which the color set of a vertex u is not contained in the color set of a neighbor of u . The least number of colors required in an inclusion-free edge-coloring is the inclusion chromatic index , denoted by χ ⊂ ′ ( G ) . In the paper, we show that if G is a bipartite graph with bipartition ( A , B ) where Δ ( A ) ≤ 3 and δ ( G ) ≥ 2 , then χ ⊂ ′ ( G ) ≤ 2 Δ + 1 .