Interval Edge Colorings Research Articles

Interval edge colorings of the generalized lexicographic product of some graphs

<p>An edge-coloring of a graph $ G $ with colors $ 1, \ldots, t $ is an interval $ t $-coloring if all colors are used and the colors of edges incident to each vertex of $ G $ are distinct and form an interval of integers. A graph $ G $ is interval colorable if it has an interval $ t $-coloring for some positive integer $ t $. For an interval colorable graph $ G $, the least and the greatest values of $ t $ for which $ G $ has an interval $ t $-coloring are denoted by $ w(G) $ and $ W(G) $. Let $ G $ be a graph with vertex set $ V(G) = \{u_1, \ldots, u_{m}\} $, $ m\geq2 $, and let $ h_m = (H_i)_{i\in\{1, \ldots, m\}} $ be a sequence of vertex-disjoint with $ V(H_i) = \{x_1^{(i)}, \ldots, x_{n_i}^{(i)}\} $, $ n_i\geq 1 $. The generalized lexicographic products $ G[h_m] $ of $ G $ and $ h_m $ is a simple graph with vertex set $ \cup_{i = 1}^{m}V(H_i) $, in which $ x_p^{(i)} $ is adjacent to $ x_q^{(j)} $ if and only if either $ u_i = u_j $ and $ x_p^{(i)}x_q^{(i)}\in E(H_i) $ or $ u_iu_j\in E(G) $. In this paper, we obtain several sufficient conditions for the generalized lexicographic product $ G[h_m] $ to have interval colorings. We also present some sharp bounds on $ w(G[h_m]) $ and $ W(G[h_m]) $ of $ G[h_m] $.</p>

ON INTERVAL EDGE-COLORINGS OF COMPLETE MULTIPARTITE GRAPHS

A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in a way that each vertex in $V_i$ is adjacent to all the other vertices in $V_j$ for $1\leq i<j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote a complete $r$-partite graph with independent sets $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an \emph{interval $t$-coloring}, if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In this paper we have obtained some results on the existence and construction of interval edge-colorings of complete $r$-partite graphs. Moreover, we have also derived an upper bound on the number of colors in interval colorings of complete multipartite graphs.

Improper interval edge colorings of graphs

A k-improper edge coloring of a graph G is a mapping α:E(G)⟶N such that at most k edges of G with a common endpoint have the same color. An improper edge coloring of a graph G is called an improper interval edge coloring if the colors of the edges incident to each vertex of G form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph G defined as the smallest k such that G has a k-improper interval edge coloring; we denote the smallest such k by μint(G). We prove upper bounds on μint(G) for general graphs G and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine μint(G) exactly for G belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer k, there exists a graph G with μint(G)=k. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring.

Interval edge-coloring: A model of curriculum scheduling

Considering the appointments that teachers plan to teach some courses for specific classes, the problem is to schedule the curriculum such that the time for each teacher is consecutive. In this work, we propose an integer linear programming model to solve consecutive interval edge-coloring of a graph. By using the proposed method, we give the interval edge colorability of some small complete multipartite graphs. Moreover, we find some new classes of complete multipartite graphs that have interval edge-colorings and disprove a conjecture proposed by Grzesik and Khachatrian (2014).

广义θ-链的区间边着色的上界

广义θ-链的区间边着色的上界

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cyclic intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. Denote by w(G) (wc(G)) and W(G) (Wc(G)) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph G, respectively. We present some new sharp bounds on w(G) and W(G) for multigraphs G satisfying various conditions. In particular, we show that if G is a 2-connected multigraph with an interval coloring, then W(G)≤1+|V(G)|2(Δ(G)−1). We also give several results towards the general conjecture that Wc(G)≤|V(G)| for any triangle-free graph G with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most 4.

Interval Edge Coloring of Sudha Grid of Hexagons, Gear and Helm Graphs

Interval Edge Coloring of Sudha Grid of Hexagons, Gear and Helm Graphs

Some results on cyclic interval edge colorings of graphs

AbstractA proper edge coloring of a graph G with colors is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree admits a cyclic interval ‐coloring if for every vertex v the degree satisfies either or . We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for ‐biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)‐biregular graphs as well as all ‐biregular () graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

Interval edge-colorings of composition of graphs

An edge-coloring of a graph G with consecutive integers c1,…,ct is called an intervalt-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The set of all interval colorable graphs is denoted by N. In 2004, Giaro and Kubale showed that if G,H∈N, then the Cartesian product of these graphs belongs to N. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the second author showed that if G,H∈N and H is a regular graph, then G[H]∈N. In this paper, we prove that if G∈N and H has an interval coloring of a special type, then G[H]∈N. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if G∈N and T is a tree, then G[T]∈N.

Note on Cyclically Interval Edge Colorings of Simple Cycles

A proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.

On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles

A proper edge t-coloring of a graph G is a coloring of its edges with colors 1,2,???,t such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each its vertex x, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. For an arbitrary simple cycle, all possible values of t are found, for which the graph has a cyclically interval t-coloring.

Interval edge-colorings of Cartesian products of graphs I

An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if $G$ has an interval $t$-coloring for some positive integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G\in \mathfrak{N}$, the least and the greatest values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$, respectively. In this paper we first show that if $G$ is an $r$-regular graph and $G\in \mathfrak{N}$, then $W(G\square P_{m})\geq W(G)+W(P_{m})+(m-1)r$ ($m\in \mathbb{N}$) and $W(G\square C_{2n})\geq W(G)+W(C_{2n})+nr$ ($n\geq 2$). Next, we investigate interval edge-colorings of grids, cylinders and tori. In particular, we prove that if $G\square H$ is planar and both factors have at least 3 vertices, then $G\square H\in \mathfrak{N}$ and $w(G\square H)\leq 6$. Finally, we confirm the first author's conjecture on the $n$-dimensional cube $Q_{n}$ and show that $Q_{n}$ has an interval $t$-coloring if and only if $n\leq t\leq \frac{n(n+1)}{2}$.

A note on upper bounds for the maximum span in interval edge-colorings of graphs

An edge-coloring of a graph G with colors 1,…,t is an intervalt-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. In 1994, Asratian and Kamalian proved that if a connected graph G admits an interval t-coloring, then t≤(diam(G)+1)(Δ(G)−1)+1, and if G is also bipartite, then this upper bound can be improved to t≤diam(G)(Δ(G)−1)+1, where Δ(G) is the maximum degree of G and diam(G) is the diameter of G. In this note, we show that these upper bounds cannot be significantly improved.

Interval edge colorings of some products of graphs

An edge co汯r楮g of a graph G w楴h co汯rs 1 ,2,...,t 楳 ca汬ed an 楮terval t-co汯r楮g 楦 for each i 2 f1,2,...,tg there 楳 at 汥ast one edge of G co汯red by i, and the co汯rs of edges 楮c楤ent to any vertex of G are d楳t楮ct and form an 楮terval of 楮tegers. A graph G 楳 楮terval co汯rab汥, 楦 there 楳 an 楮teger t � 1 for wh楣h G has an 楮terval t-co汯r楮g. Let N be the set of a汬 楮terval co汯rab汥 graphs. In 2004 Kuba汥 and G楡ro showed that 楦 G,H 2 N, then the Cartes楡n product of these graphs be汯ngs to N. A汳o, they formu污ted a s業楬ar prob汥m

A generalization of interval edge-colorings of graphs

An edge-coloring of a graph G with colors 1 , 2 , … , t is called an interval ( t , 1 ) -coloring if at least one edge of G is colored by i , i = 1 , 2 , … , t , and the colors of edges incident to each vertex of G are distinct and form an interval of integers with no more than one gap. In this paper we investigate some properties of interval ( t , 1 ) -colorings. We also determine exact values of the least and the greatest possible number of colors in such colorings for some families of graphs.

Investigation on Interval Edge-Colorings of Graphs

An edge-coloring of a simple graph G with colors 1, 2,..., t is called an interval t-coloring [3] if at least one edge of G is colored by color i, i = 1, ..., t and the edges incident with each vertex x are colored by dG(x) consecutive colors, where dG(x) is the degree of the vertex x. In this paper we investigate some properties of interval colorings and their variations. It is proved, in particular, that if a simple graph G = (V, E) without triangles has an interval t-coloring, then t ≤ |V| − 1.

Interval edge coloring of a graph with forbidden colors

This paper is complementary to Kubale (1989). We consider herein a problem of interval coloring the edges of a graph under the restriction that certain colors cannot be used for some edges. We give lower and upper bounds on the minimum number of colors required for such a coloring. Since the general problem is strongly NP-complete, we investigate its complexity in some special cases with particular reference to those that can be solved by polynomial-time algorithms.

Interval edge-coloring of caterpillars with hairs of arbitrary length

Interval edge-coloring of caterpillars with hairs of arbitrary length