scholarly journals Injective edge coloring of graphs

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6411-6423
Author(s):  
Domingos Cardoso ◽  
Orestes Cerdeira ◽  
Charles Dominicc ◽  
Pedro Cruz
Keyword(s):  
Lower Bounds ◽  
Edge Coloring ◽  
Upper And Lower Bounds ◽  
Coloring Number ◽  
Minimum Number ◽  
Np Complete

Three edges e1, e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of lengths three. An injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e1, e2 and e3 are consecutive edges in G, then c(e1)? c(e3). The injective edge coloring number ?'i(G) is the minimum number of colors permitted in such a coloring. In this paper, exact values of ?'i(G) for several classes of graphs are obtained, upper and lower bounds for ?'i(G) are introduced and it is proven that checking whether ?'i(G) = k is NP-complete.

scholarly journals From Edge-Coloring to Strong Edge-Coloring

10.37236/3529 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Valentin Borozan ◽  
Gerard Jennhwa Chang ◽  
Nathann Cohen ◽  
Shinya Fujita ◽  
Narayanan Narayanan ◽  
...  
Keyword(s):  
Set Theory ◽  
Lower Bounds ◽  
Edge Coloring ◽  
Upper And Lower Bounds ◽  
Combinatorial Set ◽  
Proper Edge Coloring ◽  
Complete Bipartite Graphs ◽  
Np Complete ◽  
Subcubic Graphs ◽  
Combinatorial Set Theory

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: $k$-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set $S(v)$ of colors used by edges incident to a vertex $v$ does not intersect $S(u)$ on more than $k$ colors when $u$ and $v$ are adjacent. We provide some sharp upper and lower bounds for $\chi'_{k\text{-int}}$ for several classes of graphs. For $l$-degenerate graphs we prove that $\chi'_{k\text{-int}}(G)\leq (l+1)\Delta -l(k-1)-1$. We improve this bound for subcubic graphs by showing that $\chi'_{2\text{-int}}(G)\leq 6$. We show that calculating $\chi'_{k\text{-int}}(K_n)$ for arbitrary values of $k$ and $n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of $n$. Furthermore, for complete bipartite graphs we prove that $\chi'_{k\text{-int}}(K_{n,m}) = \left\lceil \frac{mn}{k}\right\rceil$. Finally, we show that computing $\chi'_{k\text{-int}}(G)$ is NP-complete for every $k\geq 1$.An addendum was added to this paper on Jul 4, 2015.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

Almost Safe Group Testing with Few Tests

1994 ◽  
Vol 3 (3) ◽  
pp. 411-419
Author(s):  
Andrzej Pelc
Keyword(s):  
Lower Bounds ◽  
Group Testing ◽  
Upper And Lower Bounds ◽  
Probability Functions ◽  
Minimum Number ◽  
Location Strategies

In group testing, sets of data undergo tests that reveal if a set contains faulty data. Assuming that data items are faulty with given probability and independently of one another, we investigate small families of tests that enable us to locate correctly all faulty data with probability converging to one as the amount of data grows. Upper and lower bounds on the minimum number of such tests are established for different probability functions, and respective location strategies are constructed.

Computing the Edge-Neighbour-Scattering Number of Graphs

2013 ◽  
Vol 68 (10-11) ◽  
pp. 599-604
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Single Vertex ◽  
Number Of Components ◽  
Np Complete

A set of edges X is subverted from a graph G by removing the closed neighbourhood N[X] from G. We denote the survival subgraph by G=X. An edge-subversion strategy X is called an edge-cut strategy of G if G=X is disconnected, a single vertex, or empty. The edge-neighbour-scattering number of a graph G is defined as ENS(G) = max{ω(G/X)-|X| : X is an edge-cut strategy of G}, where w(G=X) is the number of components of G=X. This parameter can be used to measure the vulnerability of networks when some edges are failed, especially spy networks and virus-infected networks. In this paper, we prove that the problem of computing the edge-neighbour-scattering number of a graph is NP-complete and give some upper and lower bounds for this parameter.

The Thickness of the Cartesian Product of Two Graphs

2016 ◽  
Vol 59 (4) ◽  
pp. 705-720
Author(s):  
Yichao Chen ◽  
Xuluo Yin
Keyword(s):  
Lower Bounds ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Minimal Graph ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Planar Subgraphs ◽  
Proper Subgraph

AbstractThe thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t that contains no proper subgraph of thickness t. In this paper, upper and lower bounds are obtained for the thickness, t(G ⎕ H), of the Cartesian product of two graphs G and H, in terms of the thickness t(G) and t(H). Furthermore, the thickness of the Cartesian product of two planar graphs and of a t-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph K4k,4k, the thickness of the Cartesian product of two complete bipartite graphs Kn,n and Kn,n is also given for n≠4k + 1.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals Remarks on upper and lower bounds formatching sequencibility of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2091-2099
Author(s):  
Shuya Chiba ◽  
Yuji Nakano
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Hamilton Cycle ◽  
Upper And Lower Bounds ◽  
Linear Ordering ◽  
The Relationship

In 2008, Alspach [The Wonderful Walecki Construction, Bull. Inst. Combin. Appl. 52 (2008) 7-20] defined the matching sequencibility of a graph G to be the largest integer k such that there exists a linear ordering of its edges so that every k consecutive edges in the linear ordering form a matching of G, which is denoted by ms(G). In this paper, we show that every graph G of size q and maximum degree ? satisfies 1/2?q/?+1? ? ms(G) ? ?q?1/??1? by using the edge-coloring of G, and we also improve this lower bound for some particular graphs. We further discuss the relationship between the matching sequencibility and a conjecture of Seymour about the existence of the kth power of a Hamilton cycle.

scholarly journals Minimum survivable graphs with bounded distance increase

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Selma Djelloul ◽  
Mekkia Kouider
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Interconnection Networks ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Bounded Distance ◽  
Minimum Number ◽  
International Audience

International audience We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.

Representation of permutations as products of cycles of fixed length

1976 ◽  
Vol 22 (3) ◽  
pp. 321-331 ◽  
Author(s):  
Marcel Herzog ◽  
K. B. Reid
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Fixed Length ◽  
Minimum Number

AbstractWe study the problem of representing a permutation C as a product of a minimum number, fk(C), of cycles of length k. Upper and lower bounds on fk(C) are obtained and exact results are derived for k = 2, 3, 4.

scholarly journals TRIPLE CROSSING NUMBER OF KNOTS AND LINKS

2013 ◽  
Vol 22 (02) ◽  
pp. 1350006 ◽  
Author(s):  
COLIN ADAMS
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Upper And Lower Bounds ◽  
Minimum Number ◽  
Bracket Polynomial ◽  
Knots And Links

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c3(K) for a variety of knots and links. We then use c3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to cn(K).

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