scholarly journals INTERVAL EDGE-COLORINGS OF TREES WITH RESTRICTIONS ON THE EDGES

2021 ◽  
Vol 55 (2 (255)) ◽  
pp. 113-122
Author(s):  
Albert Kh. Sahakyan ◽  
Rafayel R. Kamalian
Keyword(s):  
Positive Integer ◽  
Polynomial Algorithm ◽  
Edge Coloring ◽  
Edge Colorings ◽  
Consecutive Integers

An edge-coloring of a graph $G$ with consecutive integers $c_1,\ldots,c_t$ is called an interval t-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 $t$. In this paper, we consider the case, where there are restrictions on the edges of the tree and provide a polynomial algorithm for checking interval colorability that satisfies those restrictions.

Generalized edge-colorings of weighted graphs

2016 ◽  
Vol 08 (01) ◽  
pp. 1650015
Author(s):  
Yuji Obata ◽  
Takao Nishizeki
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Edge Coloring ◽  
Efficient Algorithms ◽  
Upper And Lower Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Edge Colorings ◽  
Integer Weight

Let [Formula: see text] be a graph with a positive integer weight [Formula: see text] for each vertex [Formula: see text]. One wishes to assign each edge [Formula: see text] of [Formula: see text] a positive integer [Formula: see text] as a color so that [Formula: see text] for any vertex [Formula: see text] and any two edges [Formula: see text] and [Formula: see text] incident to [Formula: see text]. Such an assignment [Formula: see text] is called an [Formula: see text]-edge-coloring of [Formula: see text], and the maximum integer assigned to edges is called the span of [Formula: see text]. The [Formula: see text]-chromatic index of [Formula: see text] is the minimum span over all [Formula: see text]-edge-colorings of [Formula: see text]. In the paper, we present various upper and lower bounds on the [Formula: see text]-chromatic index, and obtain three efficient algorithms to find an [Formula: see text]-edge-coloring of a given graph. One of them finds an [Formula: see text]-edge-coloring with span smaller than twice the [Formula: see text]-chromatic index.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

scholarly journals On the simultaneous edge coloring of graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450049
Author(s):  
Behrooz Bagheri Gh. ◽  
Behnaz Omoomi
Keyword(s):  
Bipartite Graph ◽  
Edge Coloring ◽  
Close Relation ◽  
Bipartite Graphs ◽  
Double Cover ◽  
Edge Colorings ◽  
Graphic Sequence

A μ-simultaneous edge coloring of graph G is a set of μ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The μ-simultaneous edge coloring of bipartite graphs has a close relation with μ-way Latin trades. Mahdian et al. (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al. (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring. In this paper, the μ-simultaneous edge coloring of graphs is studied. Moreover, the properties of the extremal counterexample to the above conjecture are investigated. Also, a relation between 2-simultaneous edge coloring of a graph and a cycle double cover with certain properties is shown and using this relation, some results about 2-simultaneous edge colorable graphs are obtained.

scholarly journals List Edge-Colorings of Series-Parallel Graphs

10.37236/1474 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Martin Juvan ◽  
Bojan Mohar ◽  
Robin Thomas
Keyword(s):  
Maximum Degree ◽  
Edge Coloring ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Parallel Graph

It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph $G$ with maximum degree at most $k$, and for every collection $(L(e):e\in E(G))$ of sets, each of size at least $k$, there exists a proper edge-coloring of $G$ such that for every edge $e\in E(G)$, the color of $e$ belongs to $L(e)$.

On generalized neighbor sum distinguishing index of planar graphs

2021 ◽  
pp. 2150147
Author(s):  
Jieru Feng ◽  
Yue Wang ◽  
Jianliang Wu
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring

For a proper [Formula: see text]-edge coloring [Formula: see text] of a graph [Formula: see text], let [Formula: see text] denote the sum of the colors taken on the edges incident to the vertex [Formula: see text]. Given a positive integer [Formula: see text], the [Formula: see text]-neighbor sum distinguishing [Formula: see text]-edge coloring of G is [Formula: see text] such that for each edge [Formula: see text], [Formula: see text]. We denote the smallest integer [Formula: see text] in such coloring of [Formula: see text] by [Formula: see text]. For [Formula: see text], Wang et al. proved that [Formula: see text]. In this paper, we show that if G is a planar graph without isolated edges, then [Formula: see text], where [Formula: see text].

Large rainbow matchings in semi-strong edge-colorings of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850021
Author(s):  
Zemin Jin ◽  
Kun Ye ◽  
He Chen ◽  
Yuefang Sun
Keyword(s):  
Lower Bounds ◽  
Edge Coloring ◽  
Natural Generalization ◽  
Edge Colorings ◽  
Colored Graphs ◽  
Rainbow Matching

The lower bounds for the size of maximum rainbow matching in properly edge-colored graphs have been studied deeply during the last decades. An edge-coloring of a graph [Formula: see text] is called a strong edge-coloring if each path of length at most three is rainbow. Clearly, the strong edge-coloring is a natural generalization of the proper one. Recently, Babu et al. considered the problem in the strongly edge-colored graphs. In this paper, we introduce a semi-strong edge-coloring of graphs and consider the existence of large rainbow matchings in it.

scholarly journals On the difference of values of the kernel function at consecutive integers

2003 ◽  
Vol 2003 (67) ◽  
pp. 4249-4262
Author(s):  
Jean-Marie De Koninck ◽  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Kernel Function ◽  
Large Family ◽  
Fixed Integer ◽  
Consecutive Integers ◽  
The Difference

For each positive integern, setγ(n)=Πp|np. Given a fixed integerk≠±1, we establish that if theABC-conjecture holds, then the equationγ(n+1)−γ(n)=khas only finitely many solutions. In the particular casesk=±1, we provide a large family of solutions for each of the corresponding equations.

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