Planar Graphs Research Papers - Academia.edu (original) (raw)

The article presents original concepts and methods for 4-coloring a plane graph and proving the Four Color Theorem without the help of a computer. The Graph decomposition key concept partitions a simple cycle basis for the graph into... more

The article presents original concepts and methods for 4-coloring a plane graph and proving the Four Color Theorem without the help of a computer. The Graph decomposition key concept partitions a simple cycle basis for the graph into ``Tiers" data structures, the starting point for additional graph structural ideas. Other 3-colors border and 3-colors congruence concepts provide 3-coloring support. Interim contraction of non-significant vertices in the tiers reduces coloring predominantly to coloring of their borders. Coloring begins with the innermost tiers. The already 4-colored sub-tiers are recursively expanded up to each tier. Color congruence methods either 3-color a sub-graph of the expanded tier, or identify color conflicting vertices and resolve the conflict. Finally the entire graph becomes 4-colored. The restored non-significant vertices are assigned colors, turning the original graph into 4-colored and concluding the proof.

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that... more

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of R, denoted by C (R) is an undirected simple graph whose vertex set is the set of all proper ideals I of R such that I ̸⊆ J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2 are joined by an edge in C(R) if and only if I1 + I2 = R. In Section 2 of this article, we classify rings R such that C (R) is planar. In Section 3 of this article, we classify rings R such that C (R) is a split graph. In Section 4 of this article, we classify rings R such that C(R) is complemented and moreover, we determine the S-vertices of C (R).

A standard way to approximate the distance between any two vertices p and q on a mesh is to compute, in the associated graph, a shortest path from p to q that goes through one of k sources, which are well-chosen vertices. Precomputing the... more

A standard way to approximate the distance between any two vertices p and q on a mesh is to compute, in the associated graph, a shortest path from p to q that goes through one of k sources, which are well-chosen vertices. Precomputing the distance between each of the k sources to all vertices of the graph yields an efficient compu-tation of approximate distances between any two vertices. One standard method for choosing k sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor FFPS of approximate geodesics com-puted using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that FFPS can be bounded in terms of the minimal va...

In this paper, we propose a convention for representing non-planar graphs and their least-crossing embeddings in a canonical way. We achieve this by using state-of-the-art tools such as canonical labelling of graphs, Nauty's Graph6 string... more

In this paper, we propose a convention for representing non-planar graphs and their least-crossing embeddings in a canonical way. We achieve this by using state-of-the-art tools such as canonical labelling of graphs, Nauty's Graph6 string and combinatorial representations for planar graphs. To the best of our knowledge, this has not been done before. Besides, we implement the mentioned procedure in a SageMath language and compute embeddings for certain classes of cubic, vertex-transitive and general graphs. Our main contribution is an extension of one of the graph data sets hosted on MathDataHub, and towards extending the SageMath codebase.

Indyk and Sidiropoulos (2007) proved that any orientable graph of genus g can be probabilistically embedded into a graph of genus g−1 with constant distortion. Viewing a graph of genus g as embedded on the surface of a sphere with g... more

Indyk and Sidiropoulos (2007) proved that any orientable graph of genus g can be probabilistically embedded into a graph of genus g−1 with constant distortion. Viewing a graph of genus g as embedded on the surface of a sphere with g handles attached, Indyk and Sidiropoulos' ...

The Four Colour Conjecture is reformulated as a statement about non-divisibility of certain binomial coefficients. This reformulation opens a (hypothetical) way of proving the Four Colour Theorem by taking advantage of recent progress in... more

The Four Colour Conjecture is reformulated as a statement about non-divisibility of certain binomial coefficients. This reformulation opens a (hypothetical) way of proving the Four Colour Theorem by taking advantage of recent progress in finding closed forms for binomial summations.

Despite of many conjectures and partial results on three colorable planar graphs nal clue has not yet been discovered. Steinberg's three coloring conjecture which asserts that all planar graphs without four and ve cycles are 3-colorable... more

Despite of many conjectures and partial results on three colorable planar graphs nal clue has not yet been discovered. Steinberg's three coloring conjecture which asserts that all planar graphs without four and ve cycles are 3-colorable is the strongest among the known similar conjectures. The author's algorithmic proof of Steinberg's conjecture based on the spiral chain coloring algorithm would not lead to a breakthrough either. The reason is that there are many planar graphs with four and ve cycles with chromatic number three (see Fig.1). Similarly another result by Grünbaum-Aksenov is that every planar graph with at most three triangles is 3-colorable extended by recent result of Borodin et.al., that there are innitely many planar 4-critical graphs with exactly four triangles. In this paper we have given a general construction of planar 3-colorable and 4-critical K4-free graphs by using a class of strong quasi edges (gadgets) Hi(u, v), where c(u) = c(v), (uv) / ∈ E(Hi(u, v)) and improper quasi edges Hi(u, v), where c(u) = c(v),(uv) / ∈ E(Hi(u, v)) for all 3colorings of G. We have also use quasi triangle Hi(u, v, w) with c(u) = c(v) = c(w) and improper quasi triangle Hi(u, v, w) with c(u) = c(v) = c(w) for all 3-colorings of G. We dene also a weak quasi edge Hi(u, v) (respectively weak quasi triangle Hi(u, v, w)) as exactly two 3-colorings with c(u) = c(v) (respectively with c(u) = c(v) = c(w)) and c(u) = c(v) (respectively c(u) = c(v) = c(w)) for all(uv) / ∈ E(Hi(u, v)) (resp.