We extend the definition of the chromatic symmetric function X G to include graphs G with a vertex-weight function w : V (G) → N. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of X G .
This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting XB to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.
We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here [0,1], among n 2 agents with different demands α1, α2,. .. , αn summing to 1? When all the agents have equal demands of α1 = α2 = • • • = αn = 1/n, it is well known that there exists a fair division with n − 1 cuts, and this is optimal. For arbitrary demands on the other hand, folklore arguments from algebraic topology show that O(n log n) cuts suffice, and this has been the state of the art for decades. Here, we improve the state of affairs in two ways: we prove that disproportionate division may always be achieved with 3n − 4 cuts, and also give an effective algorithm to construct such a division. We additionally offer a topological conjecture that implies that 2n − 2 cuts suffice in general, which would be optimal.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.