Sets of r-Graphs that Color All r-Graphs

Abstract An r -regular graph is an r -graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r -graphs. An H -coloring of G is a mapping $$f:E(G) \rightarrow E(H)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such that each r adjacent edges of G are mapped to r adjacent edges of H . For every $$r\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , let $$\mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> be an inclusion-wise minimal set of connected r -graphs, such that for every connected r -graph G there is an $$H \in \mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:mrow> </mml:math> which colors G . The Petersen Coloring Conjecture states that $$\mathcal H_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> consists of the Petersen graph P . We show that if true, then this is a very exclusive situation. Our main result is that either $$\mathcal H_3 = \{P\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>P</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> or $$\mathcal H_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> is an infinite set and if $$r \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , then $$\mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> is an infinite set. In particular, for all $$r \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , $$\mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> is unique. We first characterize $$\mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> and then prove that if $$\mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> contains more than one element, then it is an infinite set. To obtain our main result we show that $$\mathcal H_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> contains the smallest r -graphs of class 2 and the smallest poorly matchable r -graphs, and we determine the smallest r -graphs of class 2.