Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. By the famous four color theorem, every planar graph admits an independent set that contains at least one quarter of its vertices. We prove that every trianglefree planar graph on n vertices with maximum degree three has an independent set with size at least 38 n. Given all maximal independent sets of a graph, find the maximum. Proof since g is claw, k 4 free and 4regular, none of the graphs g 1, g 3, g 5, g 6, g 9 in fig.
On 3chosability of triangle free planar graphs list coloring quick reminder let g be a graph and c set of colors. This was suggested and later conjectured by albertson, bollobas, and tucker. A set of vertices in g is independent if no two of them are adjacent. Drawings of planar graphs with few slopes and segments. This is almost what you want you want the graph to be 3regular, and presumably with some work you could modify the proof to get exactly what you want. Moreover, it remains nphard under substantial restrictions, for instance, for triangle free graphs 21 and for planar cubic graphs 1. Random cubic planar graphs 81 let x be the set of vertices of g not in h. The first graph in this paper shows an 88 vertex planar. Therefore, every connected cubic graph other than k 4 has an independent set of at least n3 vertices, where n is the number of vertices in the graph. When a planar graph is drawn in this way, it divides the plane into regions called faces draw, if possible, two different planar graphs with the. Independent sets in trianglefree cubic planar graphs core. An approximation algorithm for the maximum independent set. Choukhmane and franco have presented a polynomial time approximation algorithm for the maximum independent set problem in cubic planar graphs.
If g is a connected claw, k 4 free 4regular graph, then g f t for some t with t. E, where v is a nite set and graph, g e v 2 is a set of pairs of elements in v. Oct 06, 2014 read independent sets in claw, k 4 free 4regular graphs, discrete mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The set v is called the set of vertices and eis called the set of edges of g. Combinatorial map a combinatorial object that can encode plane graphs. Also shown in 12 is a family of planar graphs of girth five once again having vertices of degree four whose fractional chromatic number converges to 11 4. Counting independent sets in trianglefree graphs je cooper, dhruv mubayiy november 15, 2011 abstract ajtai, koml os, and szemer edi proved that for su ciently large t every triangle free graph with n vertices and average degree t has an independent set of size at least n 100t logt. The maximum independent set problem is that of nding an independent set of maximum size in a given graph. Independent set on h free graphs remained unknown was the p 5. Goldberg y april 4, 2009 abstract we derive new results for the performance of a simple greedy algorithm for nding large independent sets and matchings in constant degree regular graphs. Independent sets in claw, k 4 free 4regular graphs. Subcubic triangle free graphs have fractional chromatic number at most 145. We extend this by proving that the number of independent sets in.
Independent sets in trianglefree cubic planar graphs. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k. The saturation number and independent sets on fullerene graphs and triangle free cubic planar graphs are also studied in 12, 16. The authors have also conjectured 6 that the fractional chromatic number for triangle free cubic graphs is at most 14 5. Fraughnaugh and locke proved that every nvertex triangle free connected cubic graph has an independent set of cardinality at least 11n430. Parameterized complexity of the weighted independent set. Unique coloring of planar graphs a graph gis said to be uniquely k vertex colorable if there is exactly one partition of the vertices of ginto kindependent sets, and uniquely edge k colorable if there is exactly one partition of the edges of ginto kmatchings. We give an algorithm that, given a triangle free planar graph g on n vertices and an integer k. When a connected graph can be drawn without any edges crossing, it is called planar.
Independent sets in trianglefree cubic planar graphs post. This lower bound is tight for infinitely many planar graphs, and finding maximum independent sets in planar graphs is \\mathsf np\hard. The lower bound in the previous theorem relies only on the 3. The frucht graph is one of the five smallest cubic graphs without any symmetries. Counting independent sets in triangle free graphs je cooper, dhruv mubayiy november 15, 2011 abstract ajtai, koml os, and szemer edi proved that for su ciently large t every triangle free graph with n vertices and average degree t has an independent set of size at least n 100t logt. Hamiltonian path on cubic graphs, and whether closed triangle meshes are triangle strips. In the mathematical area of graph theory, a trianglefree graph is an undirected graph in which no three vertices form a triangle of edges. Trianglefree graphs may be equivalently defined as graphs with clique number. Complexity of independent set on triangle free planar cubic graphs. The size of a maximum clique in gis called the clique number of gand is denoted. Read independent sets in claw, k 4 free 4regular graphs, discrete mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The maximum independent set problem for cubic planar.
A kpath kcycle, kface refers to a path cycle, face of length k. Case western reserve university, department of computer engineering and science, cleveland, oh 44106. However, cubic k 3, 3 minor free graphs coincide with the set of cubic planar graphs. This was suggested and later conjectured by albertson, bollob as, and tucker. G is minimum k such that can be properly colored using k colors. We prove that every triangle free planar graph on n vertices with maximum degree three has an independent set with size at least 38 n. Having these, we determine the chromatic number of a. Decomposition of complete graphs into triangles and claws fu, chinmei, lin, yuanlung, lo, shuwen, and hsu, yufong, taiwanese journal of mathematics, 2014. Pdf independent sets in trianglefree cubic planar graphs.
A graph g is planar iff g does not contain k5 or k3,3 or a subdivision of k5 or k3,3 as a subgraph. Independent sets in triangle free cubic planar graphs. If e uv2eis an edge of g, then uis called adjacent to vand uis called adjacent. L ukasz kowalik abstract although deciding whether the vertices of a planar graph can be colored with three colors is nphard, the widely known gr. Vertex cover remains npcomplete even if the input graph is planar and cubic. E 1, l be an edge length function, along with two disjoint subsets a and b of the vertices, each having even size. Any maximal independent set in this graph is formed by choosing one vertex from each triangle. Suppose to the contrary that the lemma is false and. Various polynomial time approximation algorithms, that guarantee a fixed worst case ratio between the independent set size obtained to the maximum independent set size, in planar graphs. Complexity of independent set on trianglefree planar cubic.
Proposition for chromatic number and independent sets. Planar graphs with two triangles and 3colorability of chains zden ek dvo r ak bernard lidicky y january 10, 2017 abstract aksenov proved that in a planar graph g with at most one triangle, every precoloring of a 4cycle can be extended to a 3coloring of g. Packing and covering triangles in k 4 free planar graphs. Hamiltonian path on cubic graphs, and whether closed.
According to brooks theorem every connected cubic graph other than the complete graph k 4 can be colored with at most three colors. This is tight, because, as noted by fajtlowicz 2, the generalized petersen graph p 7. Approximation algorithm for maximum independent set in. The saturation number and independent sets on fullerene graphs and trianglefree cubic planar graphs are also studied in 12, 16. Subcubic trianglefree graphs have fractional chromatic. Using this, we prove the nphardness of triangle edge deletion on planar graphs, describing a reduction from vertex cover on planar 4 in general, we cannot achieve a parameterpreserving reduction from 3hitting set to triangle. In graph theory, a uniquely colorable graph is a kchromatic graph that has only one possible proper kcoloring up to permutation of the colors. It is an open problem whether every planar graph is the intersection graph of a set of segments in. Rooted cubic graphs to count labeled cubic planar simple graphs, we introduce labeled rooted cubic planar graphs. Then each triangle in gcontains at most one ownedge. Can we draw the line between the polynomial and the nphard side, in terms of the set f.
Randomized greedy algorithms for independent sets and. Pdf use acrobat to read independent sets in trianglefree cubic planar graphs with robin thomas, journal of combinatorial theory, series b. The failed zero forcing number of a graph fetcie, katherine, jacob, bonnie, and saavedra, daniel, involve. Based on the connectivity decomposition exact counting formulas and a deterministic polynomial time sampling procedure then follow from general principles 19,20,31. In this paper we give the rst polynomial time algorithm for independent set on p 5 free graphs, resolving an open problem of 3, 9, 12, 26, 31, 38, 39, 41, 42, 45. Mar 01, 2006 read independent sets in triangle free cubic planar graphs, journal of combinatorial theory, series b on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Npcomplete problems on a 3connected cubic planar graph. Skrekovski abstract a graph is kchoosable if it can be colored whenever every vertex has a list of at least k available colors. Independent sets in trianglefree cubic planar graphs 0.
Every nvertex planar triangle free graph with maximum degree at most 3 has an independent set of size at least 3 8 n. Large independent sets in trianglefree cubic graphs. The idea in the cyclically 4edgeconnected case is, we. Large independent sets in trianglefree planar graphs. The set consisting of all vertices of g belonging to d and of all edges crossing this domain is denoted by intc. Our nal lemma in this section deals with extremal planar graphs. In that same paper, albertson, bollobas and tucker conjectured that every trianglefree planar graph has an independent set.
New npcomplete problems for faces on a triangulation of the sphere are produced. Randomized greedy algorithms for independent sets and matchings in regular graphs. Our algorithm also works for the weighted version of the problem. What is the maximum circular chromatic number of a triangle free planar graphs of maximum degree 3 presently, the known maximum is 207, which is the circular chromatic number of the dodecahedron. Fine structure of 4critical triangle free graphs i. This paper shows that independent set is npcomplete for trianglefree planar graphs whose maximum degree is at most three theorem 2 with f. Prove that if a graph has exactly two vertices of odd degrees, then they are connected by a path. Wordrepresentable planar graphs include triangle free planar graphs and, more generally, 3colourable planar graphs, as well as certain face subdivisions of triangular grid graphs, and certain triangulations of gridcovered cylinder graphs. The grotzsch graph, a triangle free graph with chromatic number 4. Albertson, bollobas, and tucker conjectured in 1976 that every trianglefree cubic planar graph on v vertices has an independent set of size at least sv, for some s, with s possibly as large as 38. Observe that intc is not necessarily a graph, while c. Disjoint triangles of a cubic line graph 277 a2 degv 3 for each. The maximum independent set problem is npcomplete even when restricted to planar graphs, cubic planar graphs or triangle free graphs. This is tight, because, as noted by fajtlowicz 3, the generalized petersen graph p 7, 2 has 14 vertices, no triangles, and no independent set with size six.
That is, under what conditions on f does the maximum. An independent vertex set of a graph g is a subset of the vertices such that no two vertices in the subset represent an edge of g. A graph with exactly 3 n3 maximal independent sets is easy to construct. Introduction in 1, albertson, bollob as and tucker showed that every trianglefree 3regular graph onv vertices has. Learn vocabulary, terms, and more with flashcards, games, and other study tools.
By the jordan curve theorem, a simple closed curve cin the plane partitions its complement into a bounded open set and an unbounded open set. Clique, independent set in a graph, a set of pairwise adjacent vertices is called a clique. Model the mesh as its dual graph, where vertices are triangles, and edges join triangles that share edges, then the problem boils down to finding a hamiltonian path in the graph. In section 2, we first describe a connection between claw, k 4 free 4regular graphs and the line graphs of cubic graphs. We prove that every trianglefree planar graph onn vertices with maximum degree three has an independent set with size at least 3 8 n. Wouter cames van batenburg, jan goedgebeur, gwenael joret submitted on 28 nov 2019. Get pdf 286kget pdf 286k a maximum independent set of a graph is a set of vertices with maximum cardinality such that no pair of vertices is connected by an edge. Independent sets in triangle free cubic planar graphs with robin thomas, journal of combinatorial theory, series b. Ideally, we would like to prove that every trianglefree planar graph gof maximum degree at most three has an independent set with size at least 3 8.
We give an algorithm that, given a trianglefree planar graph g on n vertices and an integer k. Complementarily, any graph with n vertices also has at most 3 n3 maximal independent sets. The problem of finding an absolute approximation still remains npcomplete. A more general version for 3uniform hypergraphs can be found in 5. Using this result, we prove a lower bound on the independence number for a connected claw, k 4 free 4regular graph and characterize the extremal graphs attaining the bound. An approximation algorithm for the maximum independent set problem in cubic planar graphs. An independent set in gis an induced subgraph hof gthat is an empty graph. This was rst conjectured by albertson, bollob as and. Pdf large independent sets in trianglefree planar graphs. In the mathematical area of graph theory, a clique.
Counting shortest two disjoint paths in cubic planar graphs. Packing and covering triangles in k free planar graphs. They are respectively the interior and the exterior of c. In that same paper, albertson, bollobas and tucker conjectured that every trianglefree planar graph has an independent set with size strictly larger than 1 3 v. Independent set in p free graphs in polynomial time. The maximum independent set problem for cubic planar graphs. Pdf use acrobat to read a new proof of the independence ratio of trianglefree cubic graphs with robin thomas, discrete mathematics, vol 233, pp. Nonseparating independent sets of cartesian product graphs cao, fayun and ren, han. From a computational point of view this is a di cult problem, i. Graph theory notes vadim lozin institute of mathematics university of warwick 1 introduction a graph g v. Beyond planarity wouter cames van batenburg, jan goedgebeur, and gwenael joret abstract.
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