Strong Edge Coloring Research Articles

Strong Edge Coloring of Outerplane Graphs with Independent Crossings

Strong Edge Coloring of Outerplane Graphs with Independent Crossings

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.

Strongly Proper Connected Coloring of Graphs

We study a new variant of connected coloring of graphs based on the concept of strong edge coloring (every color class forms an induced matching). In particular, an edge-colored path is strongly proper if its color sequence does not contain identical terms within a distance of at most two. A strong proper connected coloring of G is the one in which every pair of vertices is joined by at least one strongly proper path. Let spc(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)$$\\end{document} denote the least number of colors needed for such coloring of a graph G. We prove that the upper bound spc(G)≤5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)\\le 5$$\\end{document} holds for any 2-connected graph G. On the other hand, we demonstrate that there are 2-connected graphs with arbitrarily large girth satisfying spc(G)≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)\\ge 4$$\\end{document}. Additionally, we prove that graphs whose cycle lengths are divisible by 3 satisfy spc(G)≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)\\le 3$$\\end{document}. We also consider briefly other connected colorings defined by various restrictions on color sequences of connecting paths. For instance, in a nonrepetitive connected coloring of G, every pair of vertices should be joined by a path whose color sequence is nonrepetitive, that is, it does not contain two adjacent identical blocks. We demonstrate that 2-connected graphs are 15-colorable, while 4-connected graphs are 6-colorable, in the connected nonrepetitive sense. A similar conclusion with a finite upper bound on the number of colors holds for a much wider variety of connected colorings corresponding to fairly general properties of sequences. We end the paper with some open problems of concrete and general nature.

On strong edge-coloring of graphs with maximum degree 5

A strong edge-coloring of a graph G is a proper edge-coloring such that any two edges with distance at most 2 receive different colors. The strong chromatic index of G, denoted by χs′(G), is the least possible number of colors in a strong edge-coloring of G. Erdős and Nešetřil conjectured that every graph G with maximum degree Δ(G) has χs′(G)≤54Δ(G)2−12Δ(G)+14 if Δ(G) is odd and χs′(G)≤54Δ(G)2 if Δ(G) is even. In this paper, we prove that if G is a graph with Δ(G)≤5 and maximum average degree less than 225, then χs′(G)≤29. Our result implies that Erdős’ conjecture holds for the case Δ(G)=5, if G has no subgraph with average degree at least 225.

Strong edge-coloring of 2-degenerate graphs

Strong edge-coloring of 2-degenerate graphs

An Analysis of the Factors Influencing the Strong Chromatic Index of Graphs Derived by Inflating a Few Common Classes of Graphs

The problem of strong edge coloring discusses assigning colors to the edges of a graph such that distinct colors are assigned to any two edges which are either adjacent to each other or are adjacent to a common edge. The least number of colors required to define a strong edge coloring of a graph is called its strong chromatic index. This problem is equivalent to the problem of assigning collision-free frequencies to the links between the elements of a wireless sensor network. In this article, we discuss a novel way of generating new graphs from existing graphs. This graph construction is known as inflating a graph. We discuss the strong chromatic index of graphs generated by inflating some common classes of graphs and graphs derived from them. In particular, we consider the cycle graph, which is symmetric in nature, and graphs such as the path graph and the star graph, which are not symmetric. Further, we analyze the factors which influence the strong chromatic index of these inflated graphs.

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.

A note on strong edge-coloring of claw-free cubic graphs

A note on strong edge-coloring of claw-free cubic graphs

Strong edge-coloring of cubic bipartite graphs: A counterexample

A strong edge-coloring φ of a graph G assigns colors to edges of G such that φ ( e 1 ) ≠ φ ( e 2 ) whenever e 1 and e 2 are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of the line graph of G . In 1990 Faudree, Schelp, Gyárfás, and Tuza conjectured that if G is a bipartite graph with maximum degree 3 and sufficiently large girth, then G has a strong edge-coloring with at most 5 colors. In 2021 this conjecture was disproved by Lužar, Mačajová, Škoviera, and Soták. Here we give an alternative construction to disprove the conjecture.

Strong edge-colorings of sparse graphs with 3Δ − 1 colors

A strong edge-coloring of a graph G is a proper edge-coloring such that every path of length 3 uses three different colors. The strong chromatic index of G, denoted by χs′(G), is the least possible number of colors in a strong edge-coloring of G. Let G be a graph, mad(G) be the maximum average degree and Δ be the maximum degree of G. In this paper, we prove that if Δ≥6 and mad(G)<238, then χs′(G)≤3Δ−1; if Δ≥7 and mad(G)<269, then χs′(G)≤3Δ−1, which partially improves the result of Choi et al. (2018) who proved that if Δ≥7 and mad(G)<3, then χs′(G)≤3Δ.

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

A strong edge coloring of a graph G is a proper edge coloring of G such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 4 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by 4.

Strong edge colorings of graphs and the covers of Kneser graphs

AbstractA proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a ‐regular graph at least colors are needed. We show that a ‐regular graph admits a strong edge coloring with colors if and only if it covers the Kneser graph . In particular, a cubic graph is strongly 5‐edge‐colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. is false.

Strong chromatic index of generalized Jahangir graphs and generalized Helm graphs

The strong edge coloring of a graph G is a proper edge coloring that assigns a different color to any two edges which are at most two edges apart. The minimum number of color classes that contribute to such a proper coloring is said to be the strong chromatic index of G. This paper defines the strong chromatic index for the generalized Jahangir graphs and the generalized Helm graphs.

Proof of a conjecture on the strong chromatic index of Halin graphs

A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index of graph G, denoted by χs′(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C. In 2018, Hu et al. (2018) posed a conjecture which says that if G=T∪C is a Halin graph other than a wheel Wn, Ne2, or Ne4, then χs′(G)≤χs′(T)+2. The bound can be achieved. In this paper, we show that the above conjecture is true. This improves the known bound χs′(G)≤χs′(T)+3 obtained by Lai et al. (2012), and extends the results on Halin graphs G with Δ(G)≤3 obtained by Lih and Liu (2012) and Δ(G)≤4 obtained by Hu et al. (2018), respectively.

Strong edge-colorings of planar graphs with small girth

A strong t-edge-coloring of a graph H is an edge coloring with t colors, in which the edges on each path of length at most 3 receive different colors. In this article, we show that: every planar graph with girth at least 8 and maximum degree D≥4 is strong (3D−2)-edge-colorable; every planar graph with girth at least 10 and maximum degree D≥5 is strong (3D−3)-edge-colorable.

Strong Edge Coloring of Generalized Petersen Graphs

A strong edge coloring of a graph G is a proper edge coloring such that every color class is an induced matching. In 2018, Yang and Wu proposed a conjecture that every generalized Petersen graph P(n,k) with k≥4 and n&gt;2k can be strong edge colored with (at most) seven colors. Although the generalized Petersen graph P(n,k) is a kind of special graph, the strong chromatic index of P(n,k) is still unknown. In this paper, we support the conjecture by showing that the strong chromatic index of every generalized Petersen graph P(n,k) with k≥4 and n&gt;2k is at most 9.

Placement Delivery Arrays From Combinations of Strong Edge Colorings

It has recently been pointed out that placement delivery arrays (PDAs) are equivalent to strong edge colorings of bipartite graphs. In this paper we consider various methods of combining two or more strong edge colorings of bipartite graphs to obtain new ones, and therefore new PDAs. Combining PDAs in certain ways also gives a framework for obtaining PDAs with more robust and flexible parameters. We investigate how the parameters of certain strong edge colorings change after being combined with others and, after comparing the parameters of the resulting PDAs with those of the initial ones, find that subpacketization levels thusly can often be improved.

Strong Chromatic Index of Generalized Polygon Snake Graphs

A strong edge-coloring of a graph G is a proper edge-coloring such that every color class induces a matching. In this paper, we discuss the strong chromatic index of a cycle on k vertices and use the result to calculate the strong chromatic index of (m,n)k-gon snake graphs which generalize the families of graphs referred to as the (m,n) triangle snake graphs and (m,n) quadrilateral snake graphs.

Strong chromatic index of [formula omitted]-free graphs

A strong edge-coloring of a graph G is a coloring of the edges of G such that each color class is an induced matching. The strong chromatic index of G is the minimum number of colors in a strong edge-coloring of G.We show that the strong chromatic index of a claw-free graph with maximum degree Δ is at most 1.125Δ2+Δ, which confirms the conjecture of Erdős and Nešetřil from 1985 for this class of graphs for Δ≥12.We also prove an upper bound of 2−1t−2Δ2 on strong chromatic index of K1,t-free graphs with maximum degree Δ for all t≥4 and give an improved result 1.625Δ2 for unit disk graphs.

List strong edge-coloring of graphs with maximum degree 4

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 χs′(G) which is the minimum number of colors that allow a strong edge-coloring of G. Erdős and Nešetřil conjectured in 1985 that the upper bound of χs′(G) is 54Δ2 when Δ is even and 14(5Δ2−2Δ+1) when Δ is odd, where Δ is the maximum degree of G. The conjecture is proved right when Δ≤3. The best known upper bound for Δ=4 is 22 (Cranston, 2006). In this paper we extend the result of Cranston to the list version, that is, we prove that when Δ=4, list strong chromatic index is at most 22.