scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

scholarly journals Prefix Block-Interchanges on Binary and Ternary Strings

2019 ◽  
Author(s):  
Md. Khaledur Rahman ◽  
M. Sohel Rahman
Keyword(s):  
Upper Bound ◽  
Genome Rearrangement ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Linear Time Algorithm ◽  
Minimum Number ◽  
Binary Strings ◽  
Better Than

AbstractThe genome rearrangement problem computes the minimum number of operations that are required to sort all elements of a permutation. A block-interchange operation exchanges two blocks of a permutation which are not necessarily adjacent and in a prefix block-interchange, one block is always the prefix of that permutation. In this paper, we focus on applying prefix block-interchanges on binary and ternary strings. We present upper bounds to group and sort a given binary/ternary string. We also provide upper bounds for a different version of the block-interchange operation which we refer to as the ‘restricted prefix block-interchange’. We observe that our obtained upper bound for restricted prefix block-interchange operations on binary strings is better than that of other genome rearrangement operations to group fully normalized binary strings. Consequently, we provide a linear-time algorithm to solve the problem of grouping binary normalized strings by restricted prefix block-interchanges. We also provide a polynomial time algorithm to group normalized ternary strings by prefix block-interchange operations. Finally, we provide a classification for ternary strings based on the required number of prefix block-interchange operations.

scholarly journals An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

scholarly journals Packing triangles in low degree graphs and indifference graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gordana Manić ◽  
Yoshiko Wakabayashi
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Set Packing ◽  
The Best Approximation ◽  
Low Degree ◽  
International Audience ◽  
Approximation Guarantee ◽  
Vertex Disjoint

International audience We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

scholarly journals Packing Three-Vertex Paths in a Subcubic Graph

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Adrian Kosowski ◽  
Michal Malafiejski ◽  
Pawel Zyliński
Keyword(s):  
Linear Time ◽  
Packing Problem ◽  
Time Algorithm ◽  
Vertex Degree ◽  
Linear Time Algorithm ◽  
Cubic Graph ◽  
Subcubic Graph ◽  
International Audience ◽  
Subcubic Graphs

International audience In our paper we consider the $P_3$-packing problem in subcubic graphs of different connectivity, improving earlier results of Kelmans and Mubayi. We show that there exists a $P_3$-packing of at least $\lceil 3n/4\rceil$ vertices in any connected subcubic graph of order $n>5$ and minimum vertex degree $\delta \geq 2$, and that this bound is tight. The proof is constructive and implied by a linear-time algorithm. We use this result to show that any $2$-connected cubic graph of order $n>8$ has a $P_3$-packing of at least $\lceil 7n/9 \rceil$ vertices.

scholarly journals Error locating: degree constraints

2021 ◽  
Author(s):  
Christopher Dennis
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Mathematical Tool ◽  
Linear Time Algorithm ◽  
Total Size ◽  
Special Cases ◽  
Locating Array ◽  
General Error

Error graphs are a useful mathematical tool for representing failing interactions in a system. This representation is used as the basis for constructing an error locating array (ELA). However, if too many errors are present in a given error graph, it may not be possible to locate all interactions. We say that a graph is locatable if an ELA can be built. Bounds on the total size of an error graph are known, bounds on the degree an error graph can have have not been considered. In this thesis we explore the maximum degree an error graph may have while still guaranteeing its locatability. We consider special cases for 3 and 4 partite error graphs as well as developing bounds on the degree of a general error graph. We describe a linear time algorithm which can be used to generate tests which have at most one failing interaction.

scholarly journals The Cycles of the Multiway Perfect Shuffle Permutation

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
John Ellis ◽  
Hongbing Fan ◽  
Jeffrey Shallit
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Equal Size ◽  
Linear Time Algorithm ◽  
Theoretic Analysis ◽  
Cycle Structure ◽  
International Audience ◽  
Perfect Shuffle

International audience The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρ _k,n: i → ki \bmod (kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((\log kn)^2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated.

scholarly journals Isomorphism of graph classes related to the circular-ones property

2013 ◽  
Vol Vol. 15 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Andrew R. Curtis ◽  
Min Chih Lin ◽  
Ross M. Mcconnell ◽  
Yahav Nussbaum ◽  
Francisco Juan Soulignac ◽  
...  
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Circular Arc ◽  
Graph Classes ◽  
Discrete Algorithms ◽  
International Audience ◽  
Related Graph ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Discrete Algorithms International audience We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.

scholarly journals A linear time algorithm for finding an Euler walk in a strongly connected 3-uniform hypergraph

2012 ◽  
Vol Vol. 14 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Zbigniew Lonc ◽  
Pawel Naroski
Keyword(s):  
Linear Time ◽  
Natural Extension ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Uniform Hypergraph ◽  
Graph Theoretic ◽  
Uniform Hypergraphs ◽  
Discrete Algorithms ◽  
Strongly Connected ◽  
International Audience

Discrete Algorithms International audience By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.

scholarly journals Computing large and small stable models

2001 ◽  
Vol 2 (1) ◽  
pp. 1-23 ◽  
Author(s):  
MIROSŁAW TRUSZCZYŃSKI
Keyword(s):  
Upper Bound ◽  
Linear Time ◽  
Logic Program ◽  
Time Algorithm ◽  
Stable Model ◽  
Linear Time Algorithm ◽  
Constant Independent ◽  
Fixed Integer ◽  
Fixed Parameter ◽  
Stable Models

In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a logic program P have a stable model of size at least [mid ]P[mid ]−k? In contrast, we show that the problem SSM (small stable models problem) to decide whether a logic program P has a stable model of size at most k is much harder. We present two algorithms for this problem but their running time is given by polynomials of order depending on k. We show that the problem SSM is fixed-parameter intractable by demonstrating that it is W[2]-hard. This result implies that it is unlikely an algorithm exists to compute stable models of size at most k that would run in time O(mc), where m is the size of the program and c is a constant independent of k. We also provide an upper bound on the fixed-parameter complexity of the problem SSM by showing that it belongs to the class W[3].

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