Proper Edge Coloring Research Articles

A Polynomial Time Algorithm to Find the Star Chromatic Index of Trees

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$ is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.

Adjacent vertex distinguishing edge coloring of the semistrong product of paths

An adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of G is called the adjacent vertex distinguishing edge chromatic number of G, and denoted by . In this paper, use the method of structural chromatic and the method of proof to the contrary and obtain the adjacent vertex distinguishing edge chromatic numbers of the semi-strong product of two paths.

Introduction to dominated edge chromatic number of a graph

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs.

Repeated Patterns in Proper Colorings

For a fixed graph $H$, what is the smallest number of colors $C$ such that there is a proper edge-coloring of the complete graph $K_n$ with $C$ colors containing no two vertex-disjoint color-isomor...

Bilangan Kromatik Graceful pada Keluarga Graf Unicyclic

Suppose $ G $ is a graph, where $ G = \{V (G), E (G) \} $. Graceful coloring is defined by $ c: V (G) \to \{1,2, ..., k \} $ which induces a proper edge coloring $ c': E (G) \to \{1,2, ..., k- 1 \}$ defined by $c'(xy)=|c(x)-c(y)|$, where $ k \geq 2 $, $ k \in N $. Coloring is said to be graceful if these 3 conditions are satisfied, namely the proper vertex color, the proper edge color, and the edge color, which are the absolute difference between the color of the accident vertex. The subgraph $H$ on that graceful coloring is smaller than the $G$. Furthermore, one of the subgraphs in the unicyclic graph family is a cycle graph. The graceful chromatic number on a graph denoted by $ \chi_g (G) $, is the optimum number of graceful colors from graph $G$. This research aims to find graceful chromatic numbers in the unicyclic graph family, namely bull graphs, net graphs, cricket graphs, caveman graphs, peach graphs, and flowerpot graphs. The results of this study indicate that $\chi_g(C_l) \geq 4$, where $C_l$ is a unicyclic graphs.

Strict Neighbor-Distinguishing Index of Subcubic 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 $$\chi '_\mathrm{snd}(G)$$ of colors in a strict neighbor-distinguishing edge coloring of G. In this paper, we prove that every connected subcubic graph G with $$\delta (G)\ge 2$$ has $$\chi '_\mathrm{snd}(G)\le 7$$ , and moreover $$\chi '_\mathrm{snd}(G)=7$$ if and only if G is a graph obtained from the graph $$K_{2,3}$$ by inserting a 2-vertex into one edge.

Measurable versions of Vizing's theorem

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively Δ and π. The “approximate” version states that, for any Borel probability measure on the edge set and any ϵ>0, we can properly colour all but ϵ-fraction of edges with Δ+π colours in a Borel way. The “measurable” version, which is our main result, states that if, additionally, the measure is invariant, then there is a measurable proper edge colouring of the whole edge set with at most Δ+π colours.

Some Results on Palette Index of Cartesian Product Graphs

Given a proper edge coloring α of a graph G, we define the palette SG(ν, α) of a vertex ν ∈ V (G) as the set of all colors appearing on edges incident to ν. The palette index š(G) of G is the minimum number of distinct palettes occurring in a proper edge coloring of G. The windmill graph Wd(n, k) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph Kk at a shared universal vertex. In this paper, we determine the bound on the palette index of Cartesian products of complete graphs and simple paths. We also consider the problem of determining the palette index of windmill graphs. In particular, we show that for any positive integers n, k ≥ 2, š(Wd(n, 2k)) = n + 1.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

Acyclic Edge Coloring of IC-planar Graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G is the least number of colors such that G has an acyclic edge coloring and denoted by Χ′a(G). An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. In this paper, it is proved that Χ′a(G) ≤ Δ(G) + 10, if G is an IC-planar graph without adjacent triangles and Χ′a(G) ≤ Δ(G) + 8, if G is a triangle-free IC-planar graph.

A note on a conjecture of star chromatic index for outerplanar graphs

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge coloring by $k$ colors. In \cite{LB}, L. Bezegov$\acute{a}$ et al. conjectured that $\chi_{st}^{'} (G )\leq \lfloor\frac{3\Delta}{2}\rfloor+1$ when $G$ is an outerplanar graph with maximum degree $\Delta \geq 3.$ In this paper we obtained that $\chi_{st}^{'}(G) \leq \Delta+6$ when $G$ is an 2-connected outerplanar graph with diameter 2 or 3. If $G$ is an 2-connected outerplanar graph with maximum degree 5, then $\chi_{st}^{'}(G) \leq 9.$

Anti-Ramsey theory on complete bipartite graphs

We consider quadruples of positive integers with and such that every proper edge-coloring of the complete bipartite graph contains a rainbow subgraph. We show that every such quadruple with and satisfies this property and find an infinite sequence where this bound is sharp. We also define and compute some new anti-Ramsey numbers.

On the convergence of the mild random walk algorithm to generate random one-factorizations of complete graphs

The complete graph K n , for n even, has a one-factorization (proper edge coloring) with n – 1 colors. In the recent contribution [Dotan M., Linial N. (2017). ArXiv:1707.00477v2], the authors raised a conjecture on the convergence of the mild random walk on the Markov chain whose nodes are the colorings of K n . The mild random walk consists in moving from a coloring C to a recoloring C′ if and only if ϕ(C′) ≤ ϕ(C), where ϕ is the potential function that takes its minimum at one-factorizations. We show the validity of such algorithm with several numerical experiments that demonstrate convergence in all cases (not just asymptotically) with polynomial cost. We prove several results on the mild random walk, we study deeply the properties of local minimum colorings, we give a detailed proof of the convergence of the algorithm for K 4 and K 6, and we raise new conjectures. We also present an alternative to the potential measure ϕ by consider the Shannon entropy, which has a strong parallelism with ϕ from the numerical standpoint.

Twin chromatic indices of some graphs with maximum degree 3

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤ k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤ k ) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by . In this paper, we determine the twin chromatic indices of circulant graphs , and some generalized Petersen graphs such as GP(3s, k), GP(m, 2), and GP(4s, l) where n ≥ 6 and n ≡ 0 (mod 4), s ≥ 1, k ≢ 0 (mod 3), m ≥ 3 and m ∉ {4, 5}, and l is odd. Moreover, we provide some sufficient conditions for a connected graph with maximum degree 3 to have twin chromatic index greater than 3.

Normal 5-edge-colorings of a family of Loupekhine snarks

In a proper edge-coloring of a cubic graph an edge uv is called poor or rich, if the set of colors of the edges incident to u and v contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. In this note, we show that some snarks constructed by using a method introduced by Loupekhine admit a normal edge-coloring with five colors. The existence of a Berge-Fulkerson Covering for a part of the snarks considered in this paper was recently proved by Manuel and Shanthi (2015). Since the existence of a normal edge-coloring with five colors implies the existence of a Berge-Fulkerson Covering, our main theorem can be viewed as a generalization of their result.

Edge precoloring extension of hypercubes

AbstractWe consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most edges of the ‐dimensional hypercube can be extended to a proper ‐edge coloring of . Additionally, we characterize which partial edge colorings of with precisely precolored edges are extendable to proper ‐edge colorings of .

Star Edge Coloring of Cactus Graphs

A star edge coloring of a graph G is a proper edge coloring of G such that no path or cycle of length 4 is bicolored. The star chromatic index of G, denoted by $$\chi ^{\prime }_{s}(G)$$ , is the minimum k such that G admits a star edge coloring with k colors. Bezegova et al. (J Graph Theory 81(1):73–82, 2016) conjectured that the star chromatic index of outerplanar graphs with maximum degree $$\Delta$$ is at most $$\left\lfloor \frac{3\Delta }{2}\right\rfloor +1$$ . In this paper, we prove this conjecture for a class of outerplanar graphs, namely Cactus graphs, wherein every edge belongs to at most one cycle.

On the super edge local antimagic total labeling of related ladder graph

A graph G is an ordered pair of sets G = (V, E), where V is a set of vertices and E is a set of edges. The concept of labeling graphs has recently gained a lot of papularity in the area of graph theory. Antimagic labeling is the interest topic to studied. The study of antimagic labeling motivated by Hartsfield and Ringel. Arumugam et.al developed the concept of antimagic labeling that is local antimagic coloring. Thus, we study the concept of local antimagic namely edge local antimagic total labeling. By an edge local antimagic total labeling, a bijection f : V(G) ∪ E(G) → {1, 2, 3, …, |V(G)| + |E(G)|} satisfying that for any two adjacent edges e 1 and e 2, wt (e 1) ≠ wt (e 2), where for e = ab ∈ G, wt (e) = f(a) + f(b) + f(ab). Thus, any edge local antimagic total labeling induces a proper edge coloring of G if each edge e is assigned the color wt (e). It is considered to be a super edge local antimagic total coloring, if the smallest labels appear in the vertices. The chromatic number of super edge local antimagic total, denoted by γleat (G), is the minimum number of colors taken over all colorings induced by super edge local antimagic total labelings of G. In this paper we study edge local antimagic total labeling and determined the chromatic number of graphs as follow grid graph, prism graph and mobious ladder graph.

Super edge local antimagic total labeling of some graph operation

Let G be a simple connected, and undirected graph. Graph G has a set of vertex denoted by V(G) and a set of edge denoted by E(G). d(v) is the degree of vertex v ∈ V(G) and Δ(G) is the maximum degree of G. A total labeling of graph G(V, E) is said to be local edge antimagic total labeling if a bijection f : V(G) ∪ E(G) → {1, 2, 3, …, |V(G)| + |E(G)|} such that for any two adjacent edges e 1 and e 2, wt (e 1) ≠ wt (e 2), where for e = uv ∈ G, wt (e) = f(u) + f(uv) + f(v). The local edge antimagic total labeling induces a proper edge coloring of G if each edge e is assigned the color wt (e). The edge local antimagic chromatic number of G denoted by γelat (G), is the minimum number of distinct color induced by edge weights over all local antimagic total labeling of G. In this paper, we determined the edge local antimagic chromatic number of Diamond Ladder graph, Pn ⊙ Pm Three Circular ladder graphht, and shack(F 2, v, n).

Progress on the Adjacent Vertex Distinguishing Edge Coloring Conjecture

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.