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### Acyclic Coloring with Few Division Vertices [chapter]

Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman, Sue Whitesides
2012 Lecture Notes in Computer Science
In this paper we prove that every triangulated plane graph 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?"  ...

### Acyclic coloring with few division vertices

Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman, Sue Whitesides
2013 Journal of Discrete Algorithms
In this paper we prove that every triangulated plane graph 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.  ...

### The Optimality of Partial Clique Covering for Index Coding [article]

Xinping Yi, Giuseppe Caire
2018 arXiv   pre-print
A digraph is said to be partially 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.  ...

### On b-acyclic chromatic number of a graph [article]

Marcin Anholcer, Sylwia Cichacz, Iztok Peterin
2022 arXiv   pre-print
,k} such that c(u)≠ c(v) for any uv∈ E(G) and the induced subgraph on 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.  ...

### Counting acyclic and strong digraphs by descents [article]

Kassie Archer, Ira M. Gessel, Christina Graves, Xuming Liang
2020 arXiv   pre-print
A descent of a labeled digraph is a directed edge (s, t) 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.  ...

### 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].  ...

### The chromatic polynomial of an unlabeled graph

P Hanlon
1985 Journal of combinatorial theory. Series B (Print)
All 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  .  ...

### Efficient Computation of Sparse Hessians Using Coloring and Automatic Differentiation

Assefaw H. Gebremedhin, Arijit Tarafdar, Alex Pothen, Andrea Walther
2009 INFORMS journal on computing
We also present new analytical results on star and 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.  ...

### A Lower Bound on Cycle-Finding in Sparse Digraphs

Xi Chen, Tim Randolph, Rocco A. Servedio, Timothy Sun
2022 ACM Transactions on Algorithms
are far from 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  ...

### Distance measures for dynamic citation networks

Michael J. Bommarito, Daniel Martin Katz, Jonathan L. Zelner, James H. Fowler
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.  ...

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

Elise Laruelle, Katia Belcram, Alain Trubuil, Jean-Christophe Palauqui, Philippe Andrey
2022 bioRxiv   pre-print
together 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 ).  ...

### The Path-Cycle Symmetric Function of a Digraph

Timothy Y. Chow
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 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 .  ...

### An Introduction to the Discharging Method via Graph Coloring [article]

Daniel W. Cranston, Douglas B. West
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 coloring as examples.  ...  All trees are acyclically 2-colorable, but trees with diameter at least 3 are not star 2-colorable.  ...  (Cranston-Erman-Škrekovski  ) 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 .  ...

### Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs [article]

Maria Axenovich, Casey Tompkins, Lea Weber
2019 arXiv   pre-print
By Forb(n, H) we denote a set of bipartite graphs 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.  ...

### An introduction to the discharging method via graph coloring

Daniel W. Cranston, Douglas B. West
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 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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