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Acyclic Coloring with Few Division Vertices
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this paper we prove that every triangulated plane graph

doi:10.1007/978-3-642-35926-2_11
fatcat:b6kdafg26zajzexo3qr45ms5ge
*with*n*vertices*has a 1-subdivision that is*acyclically*3-*colorable*(respectively, 4-*colorable*), where the number of*division**vertices*is at most ... On the other hand, we prove an 1.28n (respectively, 0.3n) lower bound on the number of*division**vertices*for*acyclic*3-*colorings*(respectively, 4-*colorings*) of triangulated planar graphs. ... n*vertices*has an*acyclic*k-*coloring*, k ∈ {3, 4},*with*at most cn*division**vertices*?" ...##
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Acyclic coloring with few division vertices

2013
*
Journal of Discrete Algorithms
*

In this paper we prove that every triangulated plane graph

doi:10.1016/j.jda.2013.08.002
fatcat:t465grxbofhl7hcik22p7kefry
*with*n*vertices*has a 1-subdivision that is*acyclically*3-*colorable*(respectively, 4-*colorable*), where the number of*division**vertices*is at most ... On the other hand, we prove an 1.28n (respectively, 0.3n) lower bound on the number of*division**vertices*for*acyclic*3-*colorings*(respectively, 4-*colorings*) of triangulated planar graphs. ... Every triangulated plane graph G*with*n ≥ 3*vertices*has a 1subdivision G*with*at most 2n−5*division**vertices*that is*acyclically*3-*colorable*. Proof. ...##
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The Optimality of Partial Clique Covering for Index Coding
[article]

2018
*
arXiv
*
pre-print

A digraph is said to be partially

arXiv:1701.06181v2
fatcat:jjr2uyq5jfhj7gki6uj2e4r2eq
*acyclic*if its sub-digraph induced by the vertex*with*maximum in-degree and its incoming neighbors in the complementary digraph is*acyclic*. ... index coding if and only if the side information digraph is partially*acyclic*. ... ACKNOWLEDGEMENT Insightful discussion*with*Hua SUN (UC Irvine) is thankfully acknowledged. ...##
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On b-acyclic chromatic number of a graph
[article]

2022
*
arXiv
*
pre-print

,k} such that c(u)≠ c(v) for any uv∈ E(G) and the induced subgraph on

arXiv:2206.06478v1
fatcat:t6jp5sjs6zfsdpysrzhohisup4
*vertices*of any two*colors*i,j∈{1,...,k} induces a forest. ... In particular, we derive A_b(G) for several known graph families, derive some bounds for A_b(G), compare A_b(G)*with*some other parameters and generalize some influential tools from b-*colorings*to*acyclic*... , where the number of*division**vertices*is not greater than 2n G −6. ...##
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Counting acyclic and strong digraphs by descents
[article]

2020
*
arXiv
*
pre-print

A descent of a labeled digraph is a directed edge (s, t)

arXiv:1909.01550v3
fatcat:i6m26f7d2vhqvicsknjre7gnci
*with*s > t. We count strong tournaments, strong digraphs, and*acyclic*digraphs by descents and edges. ... To count strong tournaments we use Eulerian generating functions and to count strong and*acyclic*digraphs we use a new type of generating function that we call a graphic Eulerian generating function. ... Values of a n (u, 1), the number of*acyclic*digraphs on n*vertices**with*u descents, for n ≤ 7. The total is the number of labeled*acyclic*digraphs on n*vertices*. ...##
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Page 2435 of Mathematical Reviews Vol. 58, Issue 4
[page]

1979
*
Mathematical Reviews
*

Author’s summary: “An

*acyclic**coloring*of the*vertices*of a graph is a regular*coloring*of its*vertices*that contains no two-*color*cycles. ... If each vertex of G has a degree that is*divisible*by 3 [by 4, by 5], then the total number of*vertices*is congruent to 0 mod 2 [2 mod 4, 2 mod 10]. ...##
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The chromatic polynomial of an unlabeled graph

1985
*
Journal of combinatorial theory. Series B (Print)
*

All

doi:10.1016/0095-8956(85)90068-1
fatcat:qwd3cbhhfbhupcdy2r4ewpxqsu
*vertices*in the same cycle of n receive the same*color*in any (TI, k)-*coloring*of g. ... ,~o;r~s x:l'")...x/=) c x(g,n,k); (4.1) n R E Yn neA(gj here yn denotes the set of all labeled graphs on n*vertices*. Recall that x(g, rc, k) is the number of ways to z-*color*g*with*k*colors*. ... Paul Butler of the University of Newcastle computed the number of unlabeled*acyclic*digraphs*with*n points for n 6 18. These results appear in [2] . ...##
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Efficient Computation of Sparse Hessians Using Coloring and Automatic Differentiation

2009
*
INFORMS journal on computing
*

We also present new analytical results on star and

doi:10.1287/ijoc.1080.0286
fatcat:2inr6naw3jd2jn6gswk2wwan6m
*acyclic**coloring*of chordal graphs. ... The*coloring*variant used in the second step depends on whether the recovery in the fourth step is direct or indirect: a direct method uses star*coloring*and an indirect method uses*acyclic**coloring*. ... Fabrice Zaoui of EDF R&D MOSSE, Clamart, France for helping*with*the experiments on the power flow problem. ...##
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A Lower Bound on Cycle-Finding in Sparse Digraphs

2022
*
ACM Transactions on Algorithms
*

are far from

doi:10.1145/3417979
fatcat:h57y56tqbjawhn4dvhtb4voasa
*acyclic*. ... in an*acyclic*graph, in G is large. ... Because the total number of*vertices*is 3N , after lifting back to the original graph we have that G is (1/60)-far from*acyclic**with*probability 1 − o N (1). □ EPOCHS AND*COLOR*REVELATION The goal of ...##
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Distance measures for dynamic citation networks

2010
*
Physica A: Statistical Mechanics and its Applications
*

*Acyclic*digraphs arise in many natural and artificial processes. Among the broader set, dynamic citation networks represent a substantively important form of

*acyclic*digraphs. ... In this article, we develop and apply our sink distance measure together

*with*the single-linkage hierarchical clustering algorithm to both a two-dimensional directed preferential attachment model as well ... Namely, many documents contribute novel ideas, but very

*few*feature zero outbound citations. ...

##
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Large-scale analysis and computer modeling reveal hidden regularities behind variability of cell division patterns in Arabidopsis thaliana embryogenesis
[article]

2022
*
bioRxiv
*
pre-print

together

doi:10.1101/2022.03.23.483962
fatcat:tdoztegi4zbdzcn7tld2leogcy
*with*mathematical and computer modeling to perform a large scale, systematic analysis of variability in*division*plane orientation. ... By contrast, much less is known about variability in cell*division*plane positioning, its origin and links*with*cell geometry, and its impact on tissue organization. ... However, a key difference*with*the external domain was that a*few*, if any, simulations reproduced the observed*divisions*(Figure 7A and Figure S9 ). ...##
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The Path-Cycle Symmetric Function of a Digraph

1996
*
Advances in Mathematics
*

polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as we shall see, the cover polynomial of a certain digraph associated to a poset P coincides

doi:10.1006/aima.1996.0018
fatcat:didkxjxifvbbdmrkmp322527xa
*with*... So suppose D is an*acyclic*digraph*with*d*vertices*and let B be its associated board. ... Thus to prove the theorem it suffices to prove that for an*acyclic*digraph*with*d*vertices*the sign of the coefficient of p d is (&1) d&1 . ...##
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An Introduction to the Discharging Method via Graph Coloring
[article]

2016
*
arXiv
*
pre-print

Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph

arXiv:1306.4434v2
fatcat:znrp36yihfcwfj2fcxy3hdaf6y
*coloring*as examples. ... All trees are*acyclically*2-*colorable*, but trees*with*diameter at least 3 are not star 2-*colorable*. ... (Cranston-Erman-Škrekovski [53] ) Prove that a cycle of length*divisible*by 3*with**vertices*whose degrees cycle repeatedly through 2, 2, 3 is reducible for 5-choosability of G 2 . ...##
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Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs
[article]

2019
*
arXiv
*
pre-print

By Forb(n, H) we denote a set of bipartite graphs

arXiv:1903.09725v1
fatcat:y5hohkwc3za2jh4hzhi3umqhmu
*with*parts of sizes n each, that do not contain H as an induced bipartite subgraph respecting the sides. ... One can easily show that h(Forb(n,H))= O(n^1-s) for a positive s if H is not strongly*acyclic*. Here, we prove that h(Forb(n, H)) is linear in n for all strongly*acyclic*graphs except for four graphs. ... In Section 2 we find linear lower bounds onh(n, H) for each of the strongly*acyclic*graphs*with**few*exceptions, thus proving Theorem 1. ...##
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An introduction to the discharging method via graph coloring

2017
*
Discrete Mathematics
*

Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph

doi:10.1016/j.disc.2016.11.022
fatcat:neapvvzudbd2th7wv5qj5kf4ea
*coloring*as examples. ... All trees are*acyclically*2-*colorable*, but trees*with*diameter at least 3 are not star 2-*colorable*. ... Exercise 4. 4 . 4 ) Prove that a cycle of length*divisible*by 3*with**vertices*whose degrees cycle repeatedly through 2, 2, 3 is reducible for 5-choosability of G 2 . ...
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