scholarly journals Asymptotic Lower Bounds on Circular Chromatic Index of Snarks

10.37236/2388 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Martin Mačaj ◽  
Ján Mazák
Keyword(s):  
Lower Bounds ◽  
Alternative Proof ◽  
Cubic Graphs ◽  
Chromatic Index ◽  
Infinite Class ◽  
Cubic Graph ◽  
Asymptotically Optimal

We prove that the circular chromatic index of a cubic graph $G$ with $2k$ vertices and chromatic index $4$ is at least $3+2/k$. This bound is (asymptotically) optimal for an infinite class of cubic graphs containing bridges. We also show that the constant $2$ in the above bound can be increased for graphs with larger girth or higher connectivity. In particular, if $G$ has girth at least $5$, its circular chromatic index is at least $3+2.5/k$. Our method gives an alternative proof that the circular chromatic index of the generalised type 1 Blanuša snark $B_m^1$ is $3+2/3m$.

Lower Bounds For Induced Forests in Cubic Graphs

1987 ◽  
Vol 30 (2) ◽  
pp. 193-199 ◽  
Author(s):  
J. A. Bondy ◽  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cubic Graphs ◽  
Cubic Graph

AbstractIf G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

ON THE AVERAGE CASE CIRCUIT DELAY OF DISJUNCTION

1995 ◽  
Vol 05 (02) ◽  
pp. 275-280 ◽  
Author(s):  
BEATE BOLLIG ◽  
MARTIN HÜHNE ◽  
STEFAN PÖLT ◽  
PETR SAVICKÝ
Keyword(s):  
Lower Bounds ◽  
Wide Class ◽  
Time Complexity ◽  
Upper And Lower Bounds ◽  
Average Case ◽  
Asymptotically Optimal ◽  
Circuit Delay ◽  
Suitable Measure

For circuits the expected delay is a suitable measure for the average case time complexity. In this paper, new upper and lower bounds on the expected delay of circuits for disjunction and conjunction are derived. The circuits presented yield asymptotically optimal expected delay for a wide class of distributions on the inputs even when the parameters of the distribution are not known in advance.

scholarly journals Characterisation Results for Steiner Triple Systems and Their Application to Edge-Colourings of Cubic Graphs

2010 ◽  
Vol 62 (2) ◽  
pp. 355-381 ◽  
Author(s):  
Daniel Král’ ◽  
Edita Máčajov´ ◽  
Attila Pór ◽  
Jean-Sébastien Sereni
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Cubic Graphs ◽  
Steiner Triple Systems ◽  
Cubic Graph ◽  
Triple Systems ◽  
Forbidden Configurations ◽  
Edge Colourings

AbstractIt is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective nonaffine point-transitive Steiner triple system S.

scholarly journals On maximum matchings in cubic graphs with a bounded number of bridge-covering paths

1987 ◽  
Vol 36 (3) ◽  
pp. 441-447
Author(s):  
Gary Chartrand ◽  
S.F. Kapoor ◽  
Ortrud R. Oellermann ◽  
Sergio Ruiz
Keyword(s):  
Disjoint Paths ◽  
Maximum Matching ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Edge Disjoint ◽  
Bounded Number

It is proved that if G is a connected cubic graph of order p all of whose bridges lie on r edge-disjoint paths of G, then every maximum matching of G contains at least P/2 − └2r/3┘ edges. Moreover, this result is shown to be best possible.

scholarly journals Encoding range minima and range top-2 queries

2014 ◽  
Vol 372 (2016) ◽  
pp. 20130131 ◽  
Author(s):  
Pooya Davoodi ◽  
Gonzalo Navarro ◽  
Rajeev Raman ◽  
S. Srinivasa Rao
Keyword(s):  
Data Structure ◽  
Lower Bounds ◽  
Query Time ◽  
Constant Time ◽  
Worst Case ◽  
Asymptotically Optimal ◽  
Random Array ◽  
Case Data ◽  
Natural Way

We consider the problem of encoding range minimum queries (RMQs): given an array A [1.. n ] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ( i , j ), which returns the index containing the smallest element in A [ i .. j ], without access to the array A at query time. We give a data structure whose space usage is 2 n + o ( n ) bits, which is asymptotically optimal for worst-case data, and answers RMQs in O (1) worst-case time. This matches the previous result of Fischer and Heun, but is obtained in a more natural way. Furthermore, our result can encode the RMQs of a random array A in 1.919 n + o ( n ) bits in expectation, which is not known to hold for Fischer and Heun’s result. We then generalize our result to the encoding range top-2 query (RT2Q) problem, which is like the encoding RMQ problem except that the query RT2Q( i , j ) returns the indices of both the smallest and second smallest elements of A [ i .. j ]. We introduce a data structure using 3.272 n + o ( n ) bits that answers RT2Qs in constant time, and also give lower bounds on the effective entropy of the RT2Q problem.

scholarly journals Cubic identity graphs and planar graphs derived from trees

1970 ◽  
Vol 11 (2) ◽  
pp. 207-215 ◽  
Author(s):  
A. T. Balaban ◽  
Roy O. Davies ◽  
Frank Harary ◽  
Anthony Hill ◽  
Roy Westwick
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Planar Graphs ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Plane Trees ◽  
Nontrivial Identity

AbstractThe smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

scholarly journals The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

2012 ◽  
Vol 22 (3) ◽  
pp. 765-778
Author(s):  
Piotr Formanowicz ◽  
Krzysztof Tanaś
Keyword(s):  
Graph Theory ◽  
Computer Networks ◽  
Assignment Problem ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Algorithmic Approach ◽  
Edge Colorings ◽  
Mathematical Properties

Abstract It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan–Raspaud conjecture.

scholarly journals Cubic Graphs with Small Independence Ratio

10.37236/7272 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Xujun Liu
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Independence Number ◽  
Regular Graphs ◽  
Cubic Graphs

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and  let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás proved the upper bound  $i(3,\infty) < \frac{6}{13}$  in 1981, and McKay improved it to $i(3,\infty) < 0.45537$in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \leqslant 0.454.$

scholarly journals Clusters in a Multigraph with Elevated Density

10.37236/4847 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Mark K. Goldberg
Keyword(s):  
Lower Bounds ◽  
Vertex Degree ◽  
Chromatic Index ◽  
Gamma Delta ◽  
Dense Sets

In this paper, we prove that in a multigraph whose density $\Gamma$ exceeds the maxiimum vertex degree $\Delta$, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with $\Gamma\gt\Delta+1$, the size of any cluster is bounded from the above by $(\Gamma-3)/(\Gamma-\Delta-1)$. Finally, we show that two well-known lower bounds for the chromatic index of a multigraph are equal.

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