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Algorithms [chapter]

2011 Graph Coloring Problems  
Jaeger's Circular Flow Conjecture Berge's Strong Path Partition Conjecture Berge's Directed Path-Conjecture Minimal Orientations of Critical Graphs Alon-Tarsi Orientations and Chromatic Number Bibliography  ...  Polynomials 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Coefficients of Chromatic Polynomials Characterization of Chromatic Polynomials Chromatic Uniqueness Chromatic Equivalence Zeros of Chromatic  ... 
doi:10.1002/9781118032497.ch10 fatcat:374tktuvgvekni4fnz3dgbytjm

The Chromatic Polynomial of a Digraph [article]

Winfried Hochstättler, Johanna Wiehe
2022 arXiv   pre-print
This decomposition will confirm the equality of our chromatic polynomial of a digraph and the chromatic polynomial of the underlying undirected graph in the case of symmetric digraphs.  ...  Furthermore we will decompose our NL-coflow polynomial, which becomes the chromatic polynomial of a digraph by multiplication with the number of colors to the number of components, examining the special  ...  Probably, the chromatic polynomial of a graph is better known than the flow polynomial.  ... 
arXiv:1911.09547v3 fatcat:jhmi3sxkjrhqtcgr6mtufluvx4

Page 5776 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews  
W. (4-LNDHB; Egham) Reciprocity and polynomial properties for even flows and potentials on directed graphs. (English summary) Combin. Probab. Comput. 3 (1994), no. 1, 1-11.  ...  95}:05097 umbral chromatic polynomials. {For the entire collection see MR 94i:05001.}  ... 

On chromatic and flow polynomial unique graphs

Yinghua Duan, Haidong Wu, Qinglin Yu
2008 Discrete Applied Mathematics  
It is known that the chromatic polynomial and flow polynomial of a graph are two important evaluations of its Tutte polynomial, both of which contain much information of the graph.  ...  Finally, we show that several classes of graphs, ladders, Möbius ladders and squares of n-cycle are determined by their chromatic polynomials and flow polynomials together.  ...  The second author's research was done while he was visiting the Center for Combinatorics at Nankai University. He is very grateful for the hospitality provided by the University.  ... 
doi:10.1016/j.dam.2007.10.010 fatcat:6fxjxx7owfh73orj6su54oih7m

Page 7177 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
For odd A, the values for even f, and odd # separately, also have polynomial dependence on f and are directing dependent.  ...  A different polynomial may be required for different directings of the same undirected graph and for each odd-PD problem there is an equivalent even- PD problem on the reversed graph.  ... 

Binomial Inequalities for Chromatic, Flow, and Tension Polynomials [article]

Matthias Beck, Emerson Leon
2021 arXiv   pre-print
For example, we show that χ^*_ j ≤χ^*_ d-j, for 0 ≤ j ≤ d / 2. Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph.  ...  A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs.  ...  As is well known, for a given graph G, the number χ G (n) of proper colorings of G using n colors evaluates to a polynomial in n, and so a natural question is: which polynomials are chromatic?  ... 
arXiv:1804.00208v5 fatcat:qodcslcyknbrhj5ov2r5dzlpjm

Graph polynomials and their applications I: The Tutte polynomial [article]

Joanna Ellis-Monaghan, Criel Merino
2008 arXiv   pre-print
These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial.  ...  We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general.  ...  Jackson [Jac03] surveys zeros of both chromatic and the flow polynomials.  ... 
arXiv:0803.3079v2 fatcat:gi6d5vkh3rdfjendtjipx3y73u

Tutte relations, TQFT, and planarity of cubic graphs [article]

Ian Agol, Vyacheslav Krushkal
2015 arXiv   pre-print
A version of the Tutte linear relation for the flow polynomial at (3-√(5))/2 is shown to give a planarity criterion for 3-connected cubic graphs.  ...  A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well.  ...  The authors would like to thank the IHES for hospitality and support (NSF grant 1002477). Ian Agol was partially supported by NSF grant DMS-1406301, and by a Simons Investigator grant.  ... 
arXiv:1512.07339v1 fatcat:2ciouelmjncp5b7gee4ndlbnza

Page 1336 of Mathematical Reviews Vol. , Issue 85d [page]

1985 Mathematical Reviews  
The chromatic polynomials of connected graphs with k edges and k — 2 vertices all of degree at least 2 are determined for k > 6.  ...  problem to a class of graphs whose edges can be covered by three cobases of the flow matroid, and the construction of the required flow for this class.  ... 

Structure of the flow and Yamada polynomials of cubic graphs [article]

Ian Agol, Vyacheslav Krushkal
2018 arXiv   pre-print
The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs.  ...  An application is given to the structure of the flow polynomial of cubic graphs at zero.  ...  We would like to thank Gordon Royle for many discussions, and also for sharing with us numerical data on graph polynomials. We also thank Kyle Miller for helpful comments. V.  ... 
arXiv:1801.00502v1 fatcat:x7xdh2brfbfghlfd5zsfrgwcbi

Page 1570 of Mathematical Reviews Vol. 56, Issue 5 [page]

1978 Mathematical Reviews  
Let G be a graph and P(A, G) its chromatic polynomial.  ...  For directed graphs without double-entry cycles (flow returns tu predecessor points only) a method is given to compute a minimal blocking point set.  ... 

Graphs whose flow polynomials have only integral roots [article]

Joseph P.S. Kung, Gordon F. Royle
2009 arXiv   pre-print
We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph.  ...  We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots.  ...  See, for example, [11] . In the context of matroids, chromatic and flow polynomials would seem to be very special cases.  ... 
arXiv:0908.0181v2 fatcat:ctooxct4pfh4zlp4codahacnoq

Graphs whose flow polynomials have only integral roots

Joseph P.S. Kung, Gordon F. Royle
2011 European journal of combinatorics (Print)  
We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph.  ...  We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots.  ...  It is a little harder to give a direct counting argument analogous to the chromatic polynomial case, but this can be done by considering flows with values in the direct product Z m 2 of m copies of the  ... 
doi:10.1016/j.ejc.2011.01.010 fatcat:pxsdxmaxlfcsvmmjmplozrshwa

Graph Polynomials: Towards a Comparative Theory (Dagstuhl Seminar 16241)

Jo Ellis-Monaghan, Andrew Goodall, Johann A. Makowsky, Iain Moffatt, Marc Herbstritt
2016 Dagstuhl Reports  
The area of graph polynomials has become in recent years incredibly active, with new applications and new graph polynomials being discovered each year.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 16241 "Graph Polynomials: Towards a Comparative Theory".  ...  Tutte constructed his dichromate as a bivariate generalization of the chromatic polynomial and flow polynomial. The Tutte polynomial extends from graphs to matroids more generally.  ... 
doi:10.4230/dagrep.6.6.26 dblp:journals/dagstuhl-reports/Ellis-MonaghanG16 fatcat:6mhgqibwa5av7dxeye2y6wgytm

Computing Tutte Polynomials

Gary Haggard, David J. Pearce, Gordon Royle
2010 ACM Transactions on Mathematical Software  
It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete  ...  The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model, is a 2-variable polynomial graph invariant of considerable importance in both combinatorics and statistical  ...  We wish to thank the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, for generous support during the programme on Combinatorics and Statistical Mechanics (January -June 2008)  ... 
doi:10.1145/1824801.1824802 fatcat:hvug6yat6ndm3dphurbbz32yne
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