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Register a new userBijective proofs of proper coloring theorems
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Authors
Bruce E. Sagan
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The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.
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Define a permutation to be any sequence of distinct positive integers. Given two permutations p and s on disjoint underlying sets, we denote by p sh s the set of shuffles of p and s (the set of all permutations obtained by interleaving the two permutations). A permutation statistic is a function St whose domain is the set of permutations such that St(p) only depends on the relative order of the elements of p. A permutation statistic is shuffle compatible if the distribution of St on p sh s depends only on St(p) and St(s) and their lengths rather than on the individual permutations themselves. This notion is implicit in the work of Stanley in his theory of P-partitions. The definition was explicitly given by Gessel and Zhuang who proved that various permutation statistics were shuffle compatible using mainly algebraic means. This work was continued by Grinberg. The purpose of the present article is to use bijective techniques to give demonstrations of shuffle compatibility. In particular, we show how a large number of permutation statistics can be shown to be shuffle compatible using a few simple bijections. Our approach also leads to a method for constructing such bijective proofs rather than having to treat each one in an ad hoc manner. Finally, we are able to prove a conjecture of Gessel and Zhuang about the shuffle compatibility of a certain statistic.
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by KH{o}nig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence theory and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge-coloring with $d + O(sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizings theorem is true, then our method will show that $d+1$ colors are always enough.
In this paper we answer a question posed by R. Stanley in his collection of Bijection Proof Problems (Problem 240). We present a bijective proof for the enumeration of walks of length $k$ a chess rook can move along on an $mtimes n$ board starting and ending on the same square.
We give a shorter proof of the following theorem of Kathryn Mann cite{M}: the identity component of the group of the compactly supported $C^r$ diffeomorphisms of $R^n$ cannot admit a nontrivial $C^p$-action on $S^1$, provided $ngeq2$, $r eq n+1$ and $pgeq2$. We also give a new proof of another theorem of Mann: any nontrivial endomorphism of the group of the orientation preserving $C^r$ diffeomorphisms of the circle is the conjugation by a $C^r$ diffeomorphism, if $rgeq3$.
Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schroder paths (resp. little Schroder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: begin{align*} r(n,0)&=sumlimits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+sumlimits_{j=k}^{n-1}2^{j-k}r(n-1,j),quad 1le kle n, s(n,0) &=sumlimits_{j=1}^{n-1}2cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+sumlimits_{j=k+1}^{n-1}2cdot3^{j-k-1}s(n-1,j),quad 1le kle n. end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,kge 0}$ and $[s(n,k)]_{n,kge 0}$, whose row sums produce the large and little Schroder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their $A$- and $Z$-sequences characterizations. On the other hand, it is well-known that the large Schroder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by $[r(n,k)]_{n,kge 0}$ as well.
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