Uniquely 4-colorable planar graphs pdf

The uniquely 4colorable planar graphs are known to be exactly the apollonian networks, that is, the planar 3trees fowler 1998 properties. If all simple 3regular 2edgeconnected plane graphs are 4face colorable, then all 3regular 2edgeconnected plane graphs are 4face colorable 7. The uniquely 4colorable planar graphs are known to be exactly the apollonian networks, that is, the planar 3trees fowler 1998. A kchromatic graph g is called uniquely kcolorable if every kcoloring of the vertex set v of g induces the same partition of v into k color classes. Different from related work mentioned above, an alternative method only involving 4 reducible unavoidable sets for proving the four color theorem is used in this paper, which takes form of. Indeed, in order to prove fourcolor conjecture, it clearly suffices to show that all maximal planar graphs are 4colorable. If there exists a unique proper coloring for g from this list of colors, then g is. The fourcolor conjecture says that every planar graph without loops is 4colorable. Pdf on unique coloring of planar graphs researchgate. There is an infinite class c of uniquely 4colorable planar graphs obtained from the k 4 by repeatedly inserting new vertices of degree 3 in triangular faces. A kcoloring of a graph g is an assignment of k colors to the vertices of g. The famous fourcolor theorem, proved in 1976, says that the vertices of any planar graph can be colored in four colors so that adjacent vertices receive different colors. Then, prove that all planar graphs without triangles are fourcolorable without using the. Stiebitz and v oigt t ec hnisc he univ ersit at ilmenau, d98684 ilmenau, german y april 15, 1998 abstract a kc hromatic graph g is called uniquely kc olor able if ev ery kcoloring of the v ertex set v of g induces the same partition of in to k color classes.

For other terminologies and notations in graph theory, we refer to 4. On uniquely 3colorable planar graphs sciencedirect. May points out that the fourcolor conjecture belongs uniquely to francis guthrie and could fairly be called guthries conjecture. A graph is planar if it can be drawn in the plane without crossings. Mingshen wu1 and weihu hong2 1department of math, stat, and computer science, university of wisconsinstout, menomonie, wi 54751. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Stiebitz and v oigt t ec hnisc he univ ersit at ilmenau, d98684 ilmenau, german y april 15, 1998 abstract a kc hromatic graph g is called uniquely kc olor.

There is an infinite class c of uniquely 4colorable planar. Is this true, i know that if a graph is planar it is four colorable, but is it true that if a graph is four colorable it must be a planar graph. Theory on structure and coloring of maximal planar graphs. Fisk, that a uniquely edge 3colorable cubic planar graph with at least four vertices always contains a triangle. Prove that every planar graph without a triangle that is, a cycle of length three has a vertex of degree three or less. On uniquely 3colorable plane graphs without adjacent. Every 4colorable graph with maximum degree 4 has an. A kchromatic graph g is called uniquely kcolorable if every kcoloring of the vertex set v of g induces the same partition of v. Pdf it is shown that the chromatic number xg k of a uniquely colorable.

The four color theorem asserts that every planar graph and therefore every map on the plane or sphere no matter how large or complex, is 4colorable. It is easy to see that there are infinitely many uniquely kcolorable graphs for any positive integer k. Show that trianglefree planar graphs are fourcolorable. A graph is four colorable if and only if it is planar. In particular, it is shown that uniquely 3colorable planar graphs with at least four points contain at least two triangles, uniquely 4colorable planar graphs. Theory on the structure and coloring of maximal planar graphs arxiv. Some properties of a uniquely kcolorable graph g with n vertices and m. The chromatic polynomial is an invariant for graphs that was introduced in 1912 by george david birkho. In this paper, we give some properties of edgecritical uniquely 3colorable planar graphs with n vertices and improve the upper bound of s i z e n given by matsumoto to 5 2 n. The uniquely 4colorable maximal planar graph conjecture states that a planar graph is uniquely 4colorable if and only if it is a recursive maximal planar graph. Matsumoto, the size of edgecritical uniquely 3colorable planar graphs, electron.

Pdf on feb 7, 2014, ibrahim cahit and others published almost uniquely chromatic 4critical and three colorable planar graphs find, read and cite all the research you need on researchgate. There are only two uniquely 3colorable planar graphs with three 3circuits and five points. This is equivalent to the statement that every uniquely vertex 4colorable planar graph has a. Size of edgecritical uniquely 3colorable planar graphs. Characterization of uniquely colorable and perfect graphs. In combinatorial mathematics, an apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. A labeled graph g with chromatic number n is called uniquely ncolorable or simply uniquely colorable if every two partitions of the point set of g into n color classes are the same. Theorem 3 gives a complete description of uniquely 3colorable planar graphs containing exactly three 3circuits, i. Edit the following would have been a better way for me to have ask the question. Fourcoloring planar graphs wolfram demonstrations project. As a function of the number of colors it counts all possible distinct vertex.