1

January 2018
SODA '18: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

**Publisher:** Society for Industrial and Applied Mathematics

**Bibliometrics**:

Citation Count: 0

Downloads (6 Weeks): 4, Downloads (12 Months): 4, Downloads (Overall): 4

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Baker [1] devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor ...

2

February 2016
European Journal of Combinatorics: Volume 52 Issue PA, February 2016

**Publisher:** Academic Press Ltd.

Let G be a subgraph-closed graph class with bounded maximum degree. We show that if G has balanced separators whose size is smaller than linear by a polynomial factor, then G has subexponential expansion. This gives a partial converse to a result of Nešetřil and Ossona de Mendez. As an ...

3

January 2016
Journal of Combinatorial Theory Series B: Volume 116 Issue C, January 2016

**Publisher:** Academic Press, Inc.

Fix g 1 . Every graph of large enough tree-width contains a g � g grid as a minor; but here we prove that every four-edge-connected graph of large enough tree-width contains a g � g grid as an immersion (and hence contains any fixed graph with maximum degree at ...

**Keywords**:
Grid minors, Graph immersion

4

January 2016
Journal of Combinatorial Theory Series B: Volume 116 Issue C, January 2016

**Publisher:** Academic Press, Inc.

Let G be a plane graph and T an even subset of its vertices. It has been conjectured that if all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. The case k = 3 is ...

**Keywords**:
Planar graphs, Edge-coloring, T-joins

5

October 2015
Journal of Graph Theory: Volume 80 Issue 2, October 2015

**Publisher:** John Wiley & Sons, Inc.

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.

**Keywords**:
planar graph, graph coloring

6

October 2014
European Journal of Combinatorics: Volume 41, October, 2014

**Publisher:** Academic Press Ltd.

By the Grunbaum-Aksenov Theorem (extending Grotzsch's Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such graphs. This answers a question of Erdos from 1990.

7

June 2014
Combinatorica: Volume 34 Issue 3, June 2014

**Publisher:** Springer-Verlag New York, Inc.

An immersion of a graph H into a graph G is a one-to-one mapping f : V ( H ) → V ( G ) and a collection of edge-disjoint paths in G , one for each edge of H , such that the path P uv corresponding to edge ...

**Keywords**:
05C35, 05C83

8

February 2014
European Journal of Combinatorics: Volume 36, February, 2014

**Publisher:** Academic Press Ltd.

Consider a graph G=(V,E) and, for each vertex

[email protected]?V, a subset @S(v) of neighbors of v. A @S-coloring is a coloring of the elements of V so that vertices appearing together in some @S(v) receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such ...

9

January 2014
Journal of Combinatorial Theory Series B: Volume 104, January, 2014

**Publisher:** Academic Press, Inc.

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if cycles of length at most four in a planar graph G are pairwise far apart, then G is 3-choosable. This is analogous to the problem ...

**Keywords**:
List coloring, Planar graphs

10

November 2013
Journal of Combinatorial Theory Series B: Volume 103 Issue 6, November, 2013

**Publisher:** Academic Press, Inc.

Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^n^^^1^^^/^^^1^^^2^/^2^0^0^0^0 distinct 3-colorings. We show that it has at least 2^n^/^2^1^2 distinct 3-colorings.

**Keywords**:
Coloring, Triangle-free planar graph, Graph

11

October 2013
Journal of the ACM (JACM): Volume 60 Issue 5, October 2013

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 4

Downloads (6 Weeks): 5, Downloads (12 Months): 22, Downloads (Overall): 210

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We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed ...

**Keywords**:
graphs with bounded expansion, tree-depth, First-order model checking

12

October 2013
Combinatorica: Volume 33 Issue 5, October 2013

**Publisher:** Springer-Verlag New York, Inc.

Given a graphic degree sequence D , let � ( D ) (respectively � ( D ), h ( D ), and H ( D )) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over ...

**Keywords**:
05C07, 05C15

13

August 2013
WADS'13: Proceedings of the 13th international conference on Algorithms and Data Structures

**Publisher:** Springer-Verlag

We present a dynamic data structure representing a graph G , which allows addition and removal of edges from G and can determine the number of appearances of a graph of a bounded size as an induced subgraph of G . The queries are answered in constant time. When the ...

**Keywords**:
sparse graphs, data structure, subgraphs

14

July 2013
European Journal of Combinatorics: Volume 34 Issue 5, July, 2013

**Publisher:** Academic Press Ltd.

The k-domination number of a graph is the minimum size of a set D such that every vertex of G is at distance at most k from D. We give a linear-time constant-factor algorithm for approximation of the k-domination number in classes of graphs with bounded expansion, which include e.g. ...

15

January 2013
SODA '13: Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms

**Publisher:** Society for Industrial and Applied Mathematics

**Bibliometrics**:

Citation Count: 0

Downloads (6 Weeks): 0, Downloads (12 Months): 6, Downloads (Overall): 28

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For any fixed surface Σ of genus g , we give an algorithm to decide whether a graph G of girth at least five embedded in Σ is colorable from an assignment of lists of size three in time O (| V ( G )|). Furthermore, we can allow a ...

16

September 2012
Journal of Combinatorial Theory Series B: Volume 102 Issue 5, September, 2012

**Publisher:** Academic Press, Inc.

It is well known that the spectral radius of a tree whose maximum degree is @D cannot exceed

[email protected] A similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed

[email protected]+10, and more generally, for all d-degenerate graphs, where the corresponding upper bound is

[email protected] Following this, ...

**Keywords**:
Degeneracy, Graph, Spectral radius

17

August 2012
European Journal of Combinatorics: Volume 33 Issue 6, August, 2012

**Publisher:** Academic Press Ltd.

A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O(n) edges. This bound is asymptotically optimal.

18

July 2012
European Journal of Combinatorics: Volume 33 Issue 5, July, 2012

**Publisher:** Academic Press Ltd.

For every k>=0, we define G"k as the class of graphs with tree-depth at most k, i.e. the class containing every graph G admitting a valid colouring @r:V(G)->{1,...,k} such that every (x,y)-path between two vertices where @r(x)

[email protected](y) contains a vertex z where @r(z)>@r(x). In this paper, we study the set ...

19

May 2012
European Journal of Combinatorics: Volume 33 Issue 4, May, 2012

**Publisher:** Academic Press Ltd.

A class of graphs G is @g-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is @g-bounded, then every class of graphs admitting a decomposition along cuts of small rank to graphs from G ...

20

December 2011
SIAM Journal on Discrete Mathematics: Volume 25 Issue 4, November 2011

**Publisher:** Society for Industrial and Applied Mathematics

A graph $G$ is $k$-choosable if $G$ can be properly colored whenever every vertex has a list of at least $k$ available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable.

**Keywords**:
choosability, graph coloring, crossing number