scholarly journals An improved bound on the largest induced forests for triangle-free planar graphs

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Lukasz Kowalik ◽  
Borut Luzar ◽  
Riste Skrekovski
Keyword(s):  
Planar Graphs ◽  
Free Graph ◽  
International Audience

International audience We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth.

scholarly journals On-line coloring of $I_s$-free graphs

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Iwona Cieslik ◽  
Marcin Kozik ◽  
Piotr Micek
Keyword(s):  
Greedy Algorithm ◽  
Planar Graphs ◽  
Greedy Algorithms ◽  
Independent Set ◽  
Vertex Coloring ◽  
Tight Bound ◽  
Free Graph ◽  
Coloring Problem ◽  
International Audience ◽  
On Line

International audience An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.

scholarly journals Strong Oriented Chromatic Number of Planar Graphs without Short Cycles

2008 ◽  
Vol Vol. 10 no. 1 ◽  
Author(s):  
Mickael Montassier ◽  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Abelian Group ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Oriented Graph ◽  
Minimal Order ◽  
Outerplanar Graphs ◽  
International Audience

International audience Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.

scholarly journals On the k-restricted structure ratio in planar and outerplanar graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Gruia Călinescu ◽  
Cristina G. Fernandes
Keyword(s):  
Approximation Algorithms ◽  
Planar Graphs ◽  
Rates Of Convergence ◽  
Simple Graph ◽  
Maximum Weight ◽  
Outerplanar Graphs ◽  
Weighted Version ◽  
International Audience ◽  
Unweighted Case

Graphs and Algorithms International audience A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1 = 2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.

scholarly journals Density of universal classes of series-parallel graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Jaroslav Nešetřil ◽  
Yared Nigussie
Keyword(s):  
Partial Order ◽  
Chromatic Number ◽  
Rational Number ◽  
Free Graph ◽  
Circular Chromatic Number ◽  
Density Result ◽  
Proper Subclass ◽  
International Audience ◽  
Universal Classes ◽  
Minor Free Graphs

International audience A class of graphs $\mathcal{C}$ ordered by the homomorphism relation is universal if every countable partial order can be embedded in $\mathcal{C}$. It was shown in [ZH] that the class $\mathcal{C_k}$ of $k$-colorable graphs, for any fixed $k≥3$, induces a universal partial order. In [HN1], a surprisingly small subclass of $\mathcal{C_3}$ which is a proper subclass of $K_4$-minor-free graphs $(\mathcal{G/K_4)}$ is shown to be universal. In another direction, a density result was given in [PZ], that for each rational number $a/b ∈[2,8/3]∪ \{3\}$, there is a $K_4$-minor-free graph with circular chromatic number equal to $a/b$. In this note we show for each rational number $a/b$ within this interval the class $\mathcal{K_{a/b}}$ of $0K_4$-minor-free graphs with circular chromatic number $a/b$ is universal if and only if $a/b ≠2$, $5/2$ or $3$. This shows yet another surprising richness of the $K_4$-minor-free class that it contains universal classes as dense as the rational numbers.

scholarly journals Annihilating random walks and perfect matchings of planar graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Massimiliano Mattera
Keyword(s):  
Partition Function ◽  
Random Walks ◽  
Mechanical Model ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Statistical Mechanical Model ◽  
International Audience ◽  
Statistical Mechanical ◽  
Annihilating Random Walks

International audience We study annihilating random walks on $\mathbb{Z}$ using techniques of P.W. Kasteleyn and $R$. Kenyonon perfect matchings of planar graphs. We obtain the asymptotic of the density of remaining particles and the partition function of the underlying statistical mechanical model.

scholarly journals Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Marthe Bonamy ◽  
Benjamin Lévêque ◽  
Alexandre Pinlou
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Total Coloring ◽  
List Total Coloring ◽  
List Edge Coloring ◽  
International Audience

International audience For planar graphs, we consider the problems of <i>list edge coloring</i> and <i>list total coloring</i>. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

scholarly journals List edge and list total colorings of planar graphs without non-induced 7-cycles

2013 ◽  
Vol Vol. 15 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Aijun Dong ◽  
Guizhen Liu ◽  
Guojun Li
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Total Chromatic Number ◽  
International Audience

Graphs and Algorithms International audience Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.

scholarly journals Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

scholarly journals Recognizing the P_4-structure of claw-free graphs and a larger graph class

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Luitpold Babel ◽  
Andreas Brandstädt ◽  
Van Bang Le
Keyword(s):  
Large Class ◽  
Polynomial Time ◽  
Uniform Hypergraph ◽  
Free Graph ◽  
Constructive Algorithm ◽  
Time Criterion ◽  
Graph Class ◽  
Vertex Set ◽  
International Audience

International audience The P_4-structure of a graph G is a hypergraph \textbfH on the same vertex set such that four vertices form a hyperedge in \textbfH whenever they induce a P_4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P_4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.

scholarly journals Fractional Coloring of Triangle-Free Planar Graphs

10.37236/4135 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Jean-Sébastien Sereni ◽  
Jan Volec
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Free Graph ◽  
Fractional Chromatic Number ◽  
Fractional Coloring

We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-3/(3n+1)$.

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