# Cai M.-C., Favaron O., Li H.'s (2,k)-Factor-Critical Graphs and Toughness PDF

By Cai M.-C., Favaron O., Li H.

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**Extra info for (2,k)-Factor-Critical Graphs and Toughness**

**Sample text**

Guy in their book “Unsolved Problems in Geometry” [CFG]. “If Problem 8 takes that long to settle [as the celebrated Four-Color Conjecture], we should know the answer by the year 2084,” write Victor Klee and Stan Wagon in their book “New and Old Unsolved Problems in Plane Geometry” [KW]. Are you ready? Here it is: What is the smallest number of colors sufficient for coloring the plane in such a way that no two points of the same color are unit distance apart? A. 1007/978-0-387-74642-5 2, C Alexander Soifer 2009 13 14 II Colored Plane This number is called the chromatic number of the plane and is denoted by .

Nelson” was and why he and his coauthors “apparently” attributed the problem to him (our conversation on the back seat of a car in Keszthely, Hungary, when we both attended Paul Erd˝os’s 80th birthday conference in August of 1993). Thus, at least seven mathematicians—a great group to be sure—were credited with creating the problem: Paul Erd˝os, Martin Gardner, Hugo Hadwiger, Frank 1 [Bru6]. A. 1 Who created the chromatic number of the plane problem? Publication Year Author(s) Problem creator(s) or source named [Gar2] [Had4] 1960 1961 “Leo Moser.

In 2003, the Russian turned Israeli mathematician Alexei Kanel-Belov communicated to me an incredibly beautiful short proof of this lower bound by the new generation of young Russian mathematicians, all his students. The proof was found by Alexei Merkov, a 10th grader from the Moscow High School 91, and communicated by Alexei Roginsky and Daniil Dimenstein in 1997 at a Moscow Pioneer Palace [Poisk]. The following is the author’s proof with my gentle modifications. Proof of the Lower Bound (A. Merkov): Assume the plane is colored in three colors, red, white and blue, but each color forbids a distance: r, w, and b respectively.

### (2,k)-Factor-Critical Graphs and Toughness by Cai M.-C., Favaron O., Li H.

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