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On the complexity of finding the chromatic number of a recursive graph II: the unbounded case

Richard Beigel, William I. Gasarch
1989 Annals of Pure and Applied Logic  
We show that the problem of determining the chromatic number of a recursive graph with a minimum number of queries to the halting set. is close$ related to the unbounded search problem.  ...  queries to a less powerful set atic number, and 8" & Y when computhope of cutting down on the number of The most substantial savings of queries occurs in the problem recursive chromatic number of a highly  ...  In this section we show that if f(n) queries suffice to solve the unbounded search problem, then f(n) queries can be used to find the chromatic number of a recursive graph; and conversely that if d(n)  ... 
doi:10.1016/0168-0072(89)90037-7 fatcat:e6wj33qzebbwld2pv2refc2lhe

On the complexity of finding the chromatic number of a recursive graph I: the bounded case

Richard Beigel, William I. Gasarch
1989 Annals of Pure and Applied Logic  
We will be concerned with finding the chromatic number of a graph when that number is a priori bounde above by a constant. Unbounded versions of problems in tlris r are studied in [ 111.  ...  In particular, (p + l)-ary search is not always optimal for finding the chromatic number of a recursive graph.  ...  Gasmh any A, to compute # (i) Recursive grdzphs 0 ii recursive graphs c odd log(c + 1) -1 WP + 1) @Wandcodd c even c eve The complexity ofjlnding the chromatic number 31 (b) Using  ... 
doi:10.1016/0168-0072(89)90029-8 fatcat:kkh3waxg55e5llsmw7cmya5jiy

Graph polynomials and their applications II: Interrelations and interpretations [article]

Joanna Ellis-Monaghan, Criel Merino
2008 arXiv   pre-print
This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for  ...  We conclude with two examples, one from biology and one from physics, that illustrate the applicability of graph polynomials in other fields.  ...  To compute P (G; x), let G be a plane graph, and let G m be its medial graph, face two-colored with the unbounded face colored white.  ... 
arXiv:0806.4699v1 fatcat:4qzf6mxhsrgqzkgcz4xm2jimau

Page 2994 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
We show that the problem of determining the chromatic number of a recursive graph, with a minimum number of queries to the halting set, is closely related to the unbounded search problem.  ...  II. The unbounded case. Ann. Pure Appl. Logic 45 (1989), no. 3, 227-246.  ... 

Independent sets in edge-clique graphs II [article]

Ching-Hao Liu, Ton Kloks, Sheung-Hung Poon
2013 arXiv   pre-print
We show that the independent set problem on edge-clique graphs of cographs. We show that the independent set problem on edge-clique graphs of graphs without odd wheels remains NP-complete.  ...  We show that edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs.  ...  For general anti-Gallai graphs the computation of the clique number and chromatic number are NP-complete. Let us mention that the recognition of anti-Gallai graphs is NP-complete.  ... 
arXiv:1206.5082v2 fatcat:om42b4o6j5b5rl62fmq4zygosu

Book announcements

1990 Discrete Applied Mathematics  
The cardinaiity of a recursively defined set. Transformation of a number of definitions into one-jamming.  ...  Chapter 3: Parity and One-in-a-box Predicates. Chapter 4: The "And/Or" Theorem. PART II: GEOMETRIC THEORY OF LINEAR INEQUALITIES. Chapter 5: vCoNNECTED: A Geometric Property with Unbounded Order.  ... 
doi:10.1016/0166-218x(90)90074-m fatcat:uhwd4fl3wvcetkvers2zklsbry

Master index to volumes 41-50

1990 Annals of Pure and Applied Logic  
infinite paths Beigel, R. and Gasarch, W.I., On the complexity of finding the chromatic number of a recursive graph I: the bounded case Beigel, R. and Gasarch, W.I., On the complexity of finding the chromatic  ...  number of a recursive graph II: the unbounded Burke, M.R. and Magidor, M., Shelah's pcf theory and its applications Chong, CT., Hyperhypersimple sets and AZ systems Chong, C.T. and Downey, R.G., Minimal  ... 
doi:10.1016/0168-0072(90)90061-6 fatcat:inapyhmvyffpzkhhdq4qje4ahi

Binary search and recursive graph problems

William I. Gasarch, Katia S. Guimarães
1997 Theoretical Computer Science  
A graph G = (V, E) is recursive if every node of G has a finite number of neighbors, and both V and E are recursive (i.e., decidable).  ...  bounded and in the unbounded case.  ...  Acknowledgements The authors would like to thank Steven Lempp for helpful discussions, and the anonymous referees for useful observations.  ... 
doi:10.1016/s0304-3975(96)00266-6 fatcat:dz3dkn7gevdjnflzulbl2z5rm4

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory [article]

Marc Roth, Philip Wellnitz
2021 arXiv   pre-print
Counting homomorphisms from a graph H into another graph G is a fundamental problem of (parameterized) counting complexity theory.  ...  Our main result is a construction based on Kneser graphs that associates every problem P in #𝖶[1] with two classes of graphs ℋ and 𝒢 such that the problem P is equivalent to the problem # HOM(ℋ→𝒢) of  ...  Acknowledgements We thank Karl Bringmann and Holger Dell for fruitful discussions and valuable feedback on early drafts of this work.  ... 
arXiv:1907.03850v2 fatcat:rg3upoulmve7jlsnloszbm22zu

Ground-state degeneracy of Potts antiferromagnets: homeomorphic classes with noncompact W boundaries

Robert Shrock, Shan-Ho Tsai
1999 Physica A: Statistical Mechanics and its Applications  
We present exact calculations of the zero-temperature partition function Z(G,q,T=0) and ground-state degeneracy W({G},q) for the q-state Potts antiferromagnet on a number of families of graphs G for which  ...  (generalizing q from Z_+ to C) the boundary B of regions of analyticity of W in the complex q plane is noncompact, passing through z=1/q=0.  ...  Of course, since for finite r, the chromatic polynomial is of finite degree (equal to the number of vertices on the graph), its zeros (i.e., the chromatic zeros of the graph) are of bounded magnitude in  ... 
doi:10.1016/s0378-4371(98)00568-8 fatcat:mobfxc5a6nhy5a2mviz2jwqt7m

Book announcements

1991 Discrete Applied Mathematics  
The MSCCC queueing discipline). Reversible queueing networks. Applications of queueing network models (Queueing network models of computer systems.  ...  Queueing network model of a distributed computer system. Queueing network models of communication networks. Queueing model of a network of workstations. Queueing model of a circuit-switched network.  ...  Fundamental set of cycles. Cut Chapter 3: Traversability. Euler circuit. Hamiltonian cycle. Chapter 4: Node Coloring. Chromatic number. Chromatic polynomial. Chapter 5: Minimum Spanning Tree.  ... 
doi:10.1016/0166-218x(91)90006-i fatcat:gwvpijrtynhrdgyral35zuaeay

Chromatic transitions in the emergence of syntax networks [article]

Bernat Corominas-Murtra, Martí Sànchez Fibla, Sergi Valverde and Ricard Solé
2018 arXiv   pre-print
Here we explore the evolution of syntax networks through language acquisition using the chromatic number, which captures the transition and provides a natural link to standard theories on syntactic structures  ...  In a more general level, we observe that the chromatic classes define independent regions of the graph, and thus, can be interpreted as the footprints of incompatibility relations, somewhat as opposed  ...  Acknowledgments We thank Complex Systems Lab members for fruitful conversations. This work was supported by the James McDonnell Foundation (BCM, SV, RSV) and the EU program FP7-ICT-270212 (MSF).  ... 
arXiv:1807.09194v1 fatcat:ihkhflmji5f6pc5c6cj3s345bq

Acyclic and star colorings of cographs

Andrew Lyons
2011 Discrete Applied Mathematics  
We also show that the acyclic chromatic number, the star chromatic number, the treewidth plus 1, and the pathwidth plus 1 are all equal for cographs.  ...  A great deal of graph-theoretical research has been conducted on acyclic and star coloring since they were introduced in the early seventies by Grünbaum [18] .  ...  The majority of this work was conducted when the author was employed by The University of Chicago and Argonne National Laboratory, supported in part by US Department of Energy under Contract DE-AC02-06CH11357  ... 
doi:10.1016/j.dam.2011.04.011 fatcat:g2uep2reuvad5ar62d3pf6tt7u

Acyclic and Star Colorings of Cographs [article]

Andrew Lyons
2011 arXiv   pre-print
We also show that the acyclic chromatic number, the star chromatic number, the treewidth plus one, and the pathwidth plus one are all equal for cographs.  ...  If the graph is given in the form of a cotree, the algorithm runs in O(n) time.  ...  The majority of this work was conducted when the author was employed by The University of Chicago and Argonne National Laboratory, supported in part by U.S.  ... 
arXiv:1103.5531v1 fatcat:kbsfhq6idfccpouvcs3jlog4hy

Disjointness graphs of short polygonal chains [article]

János Pach, Gábor Tardos, Géza Tóth
2021 arXiv   pre-print
It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G).  ...  The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint.  ...  To prove Theorem 1, we established that the shift graph S n , a triangle-free graph of unbounded chromatic number, can be obtained as the disjointness graph of V-shapes.  ... 
arXiv:2112.05991v1 fatcat:4rlll6bgqfbbjnhvhnluywryse
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