Ibrahim Cahit

Near East University, Computer, Emeritus

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I am a retired academician and electrical engineer. My research interests are in the fields of combinatorics (graph theory) and networks.
Address: 4 Buyuk Resid
Lefkosa
via Mersin 10, Turkey

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University of Waterloo

Miquel Àngel Fiol Mora

Rahnuma Nishat

Bjarne Toft

Louis H Kauffman

University of Illinois at Chicago

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Papers by Ibrahim Cahit

On Unique Coloring of Planar Graphs

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E2671

The American Mathematical Monthly, 1978

On 2-visibility drawings of non-planar graphs

Bulletin Ica, 1999

This short note shows that 2-visibility drawing of complete graphs K_n is only possible for n<... more This short note shows that 2-visibility drawing of complete graphs K_n is only possible for n<9. The 2-visibility drawing of a graph is such that vertices are all rectangles and squares and edges are all horizontal and vertical segments. The solution of this problem has an application in VLSI-design.

From the Proof of the Four Color Theorem to P=NP

Academia Letters, 2021

On the Efficient Algorithmic Solution of the Planar Graph Three Color Problem

On harmonious tree labelings

Ars Comb., 1995

Embedding Cordially Labeled Paths: A Protein Folding Problem

Mathematics eJournal, 2001

Recently the protein folding problem has been shown to be NP-complete both for the two-and three-... more Recently the protein folding problem has been shown to be NP-complete both for the two-and three-dimensional H-P models. The simplest model is that the protein is a sequence of 0s and 1s and folding entails embedding the sequence in the two-dimensional lattice. Each such folding is evaluated with a , equal to the number of pairs of 1s that are adjacent in the lattice without being adjacent in the sequence.On the otherhand cordial labeling is, basically an binary vertex labelling problem of graphs with extra global conditions e.g. vertex and edge labels balancing condition, imposed on the vertex and the induced edge labels.In this paper we have re-formulated protein folding problem for the given special (0,1)-sequences in terms of the cordiality conditions. Some preliminary results encourages us that there exits an efficient algorithm which maximize scores by embedding the cordially labelled paths into 2- and 3-dimensional lattice points. This may give a possible answer to the well-k...

Some Results on Regularable Graphs

Mathematics eJournal, 2003

A graph is called regularable if it is possible to label its edges with integers so that the sum ... more A graph is called regularable if it is possible to label its edges with integers so that the sum of the integers assigned to the edges incident to the vertices are all same, say equal to . Clearly if the given graph is regular of degree then there is no need to find an edge-assignment; simply label all its edges with 1’s.This problem is exactly the reverse problem of the irregular assignment of graphs and can be viewed as another version of magic labelling and M-cordial labelling of graphs.Particularly we have shown that a graph can never be regularable, can be regularable with only = 0 with the use of negative integers as the edge labels or with R > 0. General characterization of graphs appear to be quite difficult graph labelling problem. In this work we have also investigated regularability of a class of graphs such as wheels, fans, grids etc.

A Proof of the Erdös - Faber - Lovász Conjecture

In 1972, Erdös -Faber -Lovász (EFL) conjectured that, if H is a linear hypergraph consisting of n... more In 1972, Erdös -Faber -Lovász (EFL) conjectured that, if H is a linear hypergraph consisting of n edges of cardinality n, then it is possible to color the vertices with n colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erdös and Frankl had given an equivalent version of the same for graphs: Let G = n i=1 A i denote a graph with n complete graphs A 1 , A 2 , . . . , A n , each having exactly n vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of G is n. In this paper we give an algorithmic proof of the conjecture using the symmetric latin squares and clique degrees of the vertices of G.

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Three Colorability of an Arrangement Graph of Great Circles

Eprint Arxiv Math 0408363, Aug 26, 2004

Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a pla... more Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a planar map? An equivalent question, under a different guise, is the following: is the arrangement graph of great circles on the sphere always vertex 3-colorable?(The arrangement graph has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points.) Assume that no three circles meet at a point, so that this arrangement graph is 4-regular. In this note we have shown that all arrangement graphs defined as above are 3-colorable.

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Embedding Trees Gracefully on the Triangular Grids

As today, after almost 35 years the famous Graceful Trees Conjecture (GTC), remains widely open. ... more As today, after almost 35 years the famous Graceful Trees Conjecture (GTC), remains widely open. Several unsuccessful methods have been devised in the past to settle this tree labeling conjecture. In this paper we re-formulated it as a tree embeding problem and show the equivalance of embedding trees with at most vertex degree three onto the triangular grids with some specific properties with the orginal graceful tree labeling.

Graceful Labeling of the Rooted Complete Trees

Abstract: The recent efforts of Hrnciar and Haviar which is based on the case analysis and simple... more Abstract: The recent efforts of Hrnciar and Haviar which is based on the case analysis and simple transformations on the selected gracefully labeled starter tree results in that “all trees of diameter five are graceful”. The resulting graceful labeling often are of the type “spiral canonic” or similar which had been used by the author a decade ago in proving gracefulness of trees of diameter four and some classes of trees of diameter five. The purpose of this paper is to elaborate the canonic labeling of the trees with some cyclic permutations on the vertex labels to show that gracefulness of a class of rooted complete trees with diameter greater than five.

Algorithmic proof of Barnette's Conjecture

In this paper we have given an algorithmic proof of an long standing Barnette's conjecture (1... more In this paper we have given an algorithmic proof of an long standing Barnette's conjecture (1969) that every 3-connected bipartite cubic planar graph is hamiltonian. Our method is quite different than the known approaches and it rely on the operation of opening disjoint chambers, bu using spiral-chain like movement of the outer-cycle elastic-sticky edges of the cubic planar graph. In fact we have shown that in hamiltonicity of Barnette graph a single-chamber or double-chamber with a bridge face is enough to transform the problem into finding specific hamiltonian path in the cubic bipartite graph reduced. In the last part of the paper we have demonstrated that, if the given cubic planar graph is non-hamiltonian then the algorithm which constructs spiral-chain (or double-spiral chain) like chamber shows that except one vertex there exists (n-1)-vertex cycle.

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On the Algorithmic Proofs of the Four Color Theorem

This paper describes algorithmic proofs of the four color theorem based on spiral chains. In the ... more

The Four-Color Map Theorem: "Kempe's Fallacious Proof Repaired

A new non-computer direct algorithmic proof for the famous four color theorem based on new concep... more A new non-computer direct algorithmic proof for the famous four color theorem based on new concept spiral-chain coloring of maximal planar graphs has been proposed by the author in 2004 [6],[13]. Historical falla-cious inductive proof of Kempe have been re-considered by many math-ematicians whether it could be repaired. All attemps so far have been either modification of Kempe color switching argument or trying to show that random second-time coloring would not produce an impasse. In this note we have shown that when Kempe's argument fails by the trap of the incomplete four-coloring there is always a simple re-coloring of the nodes of a planar graph so that the undecided node colored properly. Hence our method may be considerd as an completion of fallacious Kempe's induc-tive proof. Interesting enough, when we have resolved the impasse in the four coloring of the graphs, the solution end up again with two spirals (double-spirals) Kempe chains that cover all of the nodes.

Spiral Chains: The Proofs of Tait's and Tutte's Three-Edge-Coloring Conjectures

In this paper we have shown without assuming the four color theorem of planar graphs that every (... more In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a three-edge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the four-color conjecture at the end of the 19th century. We have also shown the applicability of our method to another well-known three edge-coloring conjecture on cubic graphs. Namely Tutte's conjecture that "every 2-connected cubic graph with no Petersen minor is 3-edge colorable". Hence the conclusion of this paper implies another non-computer proof of the four color theorem by using spiral-chains in different context.

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The Big Bang Theory of Planar Graph Coloring: Solution of the Three Color Problem ( On the construction of the three colorable planar graphs)

&amp;amp;quot; Below is my draft paper headlines (planning to present at ICM 2014) The Big Ba... more &amp;amp;quot; Below is my draft paper headlines (planning to present at ICM 2014) The Big Bang Theory of Planar Graph Coloring: Solution of the Three Color Problem Soon after Big Bang blast: 1. All graphs are triangulated and have chromatic number 4 2. Some triangulated planar disk graphs are “even” triangulated and hence with chromatic number 3. 3. Triangulated planar graphs with holes. If all holes satisfy parity symmetric conditions then the graph is 3 colorable otherwise 4 colorable by 4CT. Here PS property is violated only there is a “crash” of triangulated chain with an edge. 4. Planar graphs formed by quasi uniquely (x,x)- and (x,y) gadgets and chromatic number is 3 only if there is no crash between any two gadgets. 5. Planar graph with weakly connected triangles e.g., two triangles with common vertex. Such a planar graph has chromatic number 4 for all three colorings iff there is an non-triangle edge or vertex crashed with Kempe-tangling. 6. Planar graphs with disjoint triangles such that any two triangle apart each other by the distance at most 4. Such planar graphs have chromatic number 4 iff there is crash with quasi edges of an induce K_4. Otherwise the chromatic number is 3. 7. Planar graphs without triangles are all three colorable. 8. Planar bipartite graphs have chromatic number 2. 9. Trees are two colorable. 10. Graphs with no edges are mono-chromatic (all disjoint vertices colored by RED!) 11. End of the time (nothing to color) in the Big Bang Theory of Planar Graph Coloring. Theorem: A planar graph is 3 colorable iff there is no “crash” of two induce quasi uniquely three colorable subgraphs. &amp;amp;quot;

On the construction of three colorable planar graphs

Embedding light-paths into ring-and-mesh WDM networks

Quality of Service over Next-Generation Data Networks, 2001

Embedding light-paths into ring-and-mesh WDM networks. [Proceedings of SPIE 4524, 99 (2001)]. Ibr... more

<title>Graceful label numbering in optical MPLS networks</title>

OptiComm 2000: Optical Networking and Communications, 2000

This paper explores the positive effects of the new multi protocol label switching (MPLS) routing... more This paper explores the positive effects of the new multi protocol label switching (MPLS) routing platform in IP networks. In particular, novel node numbering algorithms based upon graceful numbering of trees are presented. The first part presents the application of the well-known graceful numbering of spanning caterpillars to the MPLS multicast routing problem. In the second part of the paper, the numbering algorithm is adjusted for the case of unicast routing in the framework of IP-over-WDM optical networks using MPLS, e.g., particularly lambda-labeling and multi protocol lambda switching.

On Unique Coloring of Planar Graphs

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E2671

The American Mathematical Monthly, 1978

On 2-visibility drawings of non-planar graphs

Bulletin Ica, 1999

From the Proof of the Four Color Theorem to P=NP

Academia Letters, 2021

On the Efficient Algorithmic Solution of the Planar Graph Three Color Problem

On harmonious tree labelings

Ars Comb., 1995

Embedding Cordially Labeled Paths: A Protein Folding Problem

Mathematics eJournal, 2001

Some Results on Regularable Graphs

Mathematics eJournal, 2003

A Proof of the Erdös - Faber - Lovász Conjecture

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Three Colorability of an Arrangement Graph of Great Circles

Eprint Arxiv Math 0408363, Aug 26, 2004

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Embedding Trees Gracefully on the Triangular Grids

Graceful Labeling of the Rooted Complete Trees

Algorithmic proof of Barnette's Conjecture

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On the Algorithmic Proofs of the Four Color Theorem

This paper describes algorithmic proofs of the four color theorem based on spiral chains. In the ... more

The Four-Color Map Theorem: "Kempe's Fallacious Proof Repaired

Spiral Chains: The Proofs of Tait's and Tutte's Three-Edge-Coloring Conjectures

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The Big Bang Theory of Planar Graph Coloring: Solution of the Three Color Problem ( On the construction of the three colorable planar graphs)

On the construction of three colorable planar graphs

Embedding light-paths into ring-and-mesh WDM networks

Quality of Service over Next-Generation Data Networks, 2001

Embedding light-paths into ring-and-mesh WDM networks. [Proceedings of SPIE 4524, 99 (2001)]. Ibr... more

<title>Graceful label numbering in optical MPLS networks</title>

OptiComm 2000: Optical Networking and Communications, 2000

The Four-Color Map Theorem: "Kempe’s Fallacious Proof Repaired"

A new non-computer direct algorithmic proof for the famous four color theorem based on new concep... more A new non-computer direct algorithmic proof for the famous four color theorem based on new concept spiral-chain coloring of maximal planar graphs has been proposed by the author in 2004 [I. Cahit, Spiral chains: A new proof of the four color theorem, ICM 2006, Madrid, Spain,2006]. Historical fallacious inductive proof of Kempe have been re-considered by many mathematicians whether it could be repaired. All attempts so far have been either modiﬁcation of Kempe color switching argument or trying to show that random second-time coloring would not produce an impasse. In this note we have shown that when Kempe’s argument fails by the trap of the incomplete four-coloring there is always a simple re-coloring of the nodes of a planar graph so that the undecided node colored properly. Hence our method may be considered as an completion of fallacious Kempe’s inductive proof. Interesting enough, when we have resolved the impasse in the incomplete four coloring of the (bad)graphs, the solution end up again with two spirals(double-spiral) Kempe-chains that cover all of the nodes

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Conjectures and Challenges in Graph Labelings

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Generalized Label Numbering for GMPLS in DWDM Networks

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Spiral Chains: The Four Color Theorem and Beyond

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On Graceful Labeling of Trees

Hrnciar and Haviar have shown that all trees of diameter five are graceful. The resulting gracef... more Hrnciar and Haviar have shown that all trees of diameter five are graceful. The resulting graceful labeling often are of the type "spiral canonic" or similar which had been used by the author in proving gracefulness of trees of diameter four and some classes of trees of diameter five. The purpose of this talk is to elaborate the canonic labeling of the trees with some cyclic permutations on the vertex labels to show that gracefulness of a class of rooted complete trees with diameter greater than five.

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A Victorian Age Proof of the Four Color Theorem

"Abstract. In this paper we have given an algorithmic proof of the four color theorem which is ba... more "Abstract. In this paper we have given an algorithmic proof of the four color theorem which is based
only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given
in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic
coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has
been obtained almost trivially."

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The Four-Color Map Theorem: "Kempe's Fallacious Proof Repaired"

A new non-computer direct algorithmic proof for the famous four color theorem based on new concep... more A new non-computer direct algorithmic proof for the famous four color theorem based on new concept spiral-chain coloring of maximal planar graphs has been proposed by the author in 2004 [6],[13]. Historical fallacious inductive proof of Kempe have been reconsidered by many mathematicians whether it could be repaired. All attemps so far have been either modification of Kempe color switching argument or trying to show that random second-time coloring would not produce an impasse. In this note we have shown that when Kempe's argument fails by the trap of the incomplete four-coloring there is always a simple re-coloring of the nodes of a planar graph so that the undecided node colored properly. Hence our method may be considerd as an completion of fallacious Kempe's inductive proof. Interesting enough, when we have resolved the impasse in the four coloring of the graphs, the solution end up again with two spirals (double-spirals) Kempe chains that cover all of the nodes.

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A Victorian Age Proof of the Four Color Theorem

In this paper we have investigated some old issues concerning four color map problem. We have giv... more In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counterexamples to Kempe's proof of the four color theorem and then show that all counterexamples can be rule out by reconstructing special 2-colored two paths decomposition in the form of a double-spiral chain of the maximal planar graph. In the second part of the paper we have given an algorithmic proof of the four color theorem which is based only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has been obtained almost trivially.

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Visualization of the Four Color Theorem

This paper describes the role of spiral-chains in the solution of some graph coloring problems in... more This paper describes the role of spiral-chains in the solution of some graph coloring problems in the theory of graphs including the recent non-computer proof of the famous four color theorem given by the author. Throughout the paper particular attention will be given on the visualization of the proof and the reasons behind the spiral chain coloring that makes the solution of the problem simple and readable.

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Embedding cubic trees into the rectangular grids

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Embedding Cubic Trees into the Rectangular Grids

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From the Proof of the Four Color Theorem

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Almost Uniquely Chromatic 4-Critical and Three Colorable Planar Graphs

Despite of many conjectures and partial results on three colorable planar graphs nal clue has not... 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.

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Embedding Trees Gracefully on the Triangular Grids

wr wkh phpru| ri Vhujlr Uxl} Fruqhmr ,ALMK@BM Dv wrgd|/ diwhu doprvw 68 |hduv wkh idprxv Judfhixo... more wr wkh phpru| ri Vhujlr Uxl} Fruqhmr ,ALMK@BM Dv wrgd|/ diwhu doprvw 68 |hduv wkh idprxv Judfhixo Wuhhv Frq0 mhfwxuh +JWF,/ uhpdlqv zlgho| rshq1 Vhyhudo xqvxffhvvixo phwkrgv kdyh ehhq ghylvhg lq wkh sdvw wr vhwwoh wklv wuhh odeholqj frqmhfwxuh1 Lq wklv sdshu zh uh0irupxodwhg lw dv d wuhh hpehglqj sureohp dqg vkrz wkh htxlydodqfh ri hp0 ehgglqj wuhhv zlwk dw prvw yhuwh{ ghjuhh wkuhh rqwr wkh wuldqjxodu julgv zlwk vrph vshflf surshuwlhv zlwk wkh rujlqdo judfhixo wuhh odeholqj1 41 Lqwurgxfwlrq

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Almost Uniquely Chromatic 4-Critical Planar Graphs

Despite of many conjectures and partial results on three colorable planar graphs nal clue has not... 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 forbidden 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 forbidden 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.

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A Short Proof of Groetzsch's Three Color Theorem

We have given a short and efficient algorithmic proof of a theorem of Grötzsch that all triangle-... more

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On the Algorithmic Proofs of the Four Color Theorem

This paper describes algorithmic proofs of the four color theorem based on spiral chains. In Aug... more This paper describes algorithmic proofs of the four color theorem based on spiral chains.
In August 2004 the author has introduced spiral chains for the maximal planar graphs in the sequel of a non-computer proof of the famous four color theorem. A year later he has given an independent proof of the equivalent problem of three-edge coloring of bridgeless cubic planar graphs which is known as Tait’s reduction, also by the use of spiral-chains. In this paper we have summarized the three proof techniques to four color theorem.

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A Short Proof of Groetzsch's Three Color Theorem

We have given a short and efficient algorithmic proof of a theorem of Grötzsch that all triangle-... more

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A Short Proof of Groetzsch's Three Color Theorem

We have given a short and efficient algorithmic proof of a theorem of Grötzsch that all triangle-... more

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A Short Proof of Groetzsch's Three Color Theorem

We have given a short and efficient algorithmic proof of a theorem of Grötzsch that all triangle-... more

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A Short Proof of Groetzsch

We have given a short and efficient algorithmic proof of a theorem of Grötzsch that all triangle-... more

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Embedding Cubic Trees into the Rectangular Grids

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The proof of the Erd®sFaberLovász conjecture

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Embedding Trees Gracefully on the Triangular Grids

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From the Proof of the Four Color Theorem to P=NP

Computer assisted proofs of the four color theorem are all alike and complexity of the coloring a... more Computer assisted proofs of the four color theorem are all alike and complexity of the coloring algorithm used in the proof is not better than a quadratic-time algorithm, improving on a quartic-time algorithm based on Appel and Haken's proof (1976) (Thomas 1995; Robertson et al. 1996). On the other hand three years ago from the Appel and Haken's proof Stockmeyer without mentioning to the four color conjecture has shown that the problem of determining whether a planar graph is three-colorable is NP-complete (Stockmeyer 1973). Thus putting an barrier on the possible efficient solution of the pla-nar graph three colorability problem although the case for maximal planar graphs has been settled by a simple necessary and sufficient condition by Heawood in 1890 and the case of triangle-free planar graphs has been settled by an rather complicated way by Groestzsch in 1958. Algorithmic proofs of the four color theorem which are all based on the spiral chains decomposition of planar graphs and normal maps with a spiral chain coloring algorithm is much simpler than the other computer assisted proofs (Cahit 2006). Despite of many conjectures and partial results on three colorable planar graphs final clue has not yet been discovered till the introduction of the crashes in planar graphs (Cahit 2014). In this paper by classification of planar graphs into three classes based on positioning and existence of the triangles in the graph and modification of the spiral chain coloring algorithm an simple and efficient algorithmic solution is given for the three color pla-nar graph problem. Steinberg's three coloring conjecture which asserts that all planar graphs without four and five 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 [I. Cahit, The proof of Steinberg's three coloring conjecture, arXiv:math/0607497 [math.CO], July 2006]. The reason is that there are many planar graphs with four and five cycles with chromatic number three. 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., [Oleg V. Borodin, et. al., Planar 4-critical graphs with four triangles, arXiv:1306.1477v1 [math.CO], June 2013] that there are infinitely many planar 4-critical graphs with exactly four triangles. Based on the use of quasi edges and a quasi triangle a class of almost uniquely 3-colorable planar graphs can be obtained easily. Similarly using forbidden quasi edges and triangles almost uniquely 4-critical planar graphs can be obtained. In the construction of these planar graphs the operations used among the quasi edges (triangles), forbidden quasi edges (triangle) are called color match (c-match), color crash (c-crash) and color trap (c-trap) in case of an triangle. Based on the above arguments we have given the following theorem:

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On the Efficient Algorithmic Solution of the Planar Graph Three Color Problemm

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A Unified Spiral Chain Coloring Algorithm for Planar Graphs

In this paper we have given a unified graph coloring algorithm for planar graphs. The problems th... more In this paper we have given a unified graph coloring algorithm for planar graphs. The problems that have been considered in this context respectively, are vertex, edge, total and entire colorings of the planar graphs. The main tool in the coloring algorithm is the use of spiral chain which has been used in the non-computer proof of the four color theorem in 2004. A more precies explanation of the proof of the four color theorem by spiral chain coloring is also given in this paper. Then we continue to spiral-chain coloring solutions by giving the proof of other famous conjectures of Vizing's total coloring and planar graph conjectures of maximum vertex degree six. We have also given the proof of a conjecture of Kronk and Mitchem that any plane graph of maximum degree ∆ is entirely (∆ + 4)-colorable. The last part of the paper deals with the three colorability of planar graphs under the spiral chain coloring. We have given an efficient and short proof of the Grötzsch's Theorem that triangle-free planar graphs are 3-colorable.

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