Combinatorics (math.CO)

Abstract: We study rotation $r$-graphs and show that for every $r$-graph $G$ of odd regularity there is a simple rotation $r$-graph $G'$ such that $G$ can be obtained form $G'$ by a finite number of $2$-cut reductions. As a consequence, some hard conjectures as the (generalized) Berge-Fulkerson Conjecture and Tutte's 3- and 5-flow conjecture can be reduced to rotation $r$-graphs.

Abstract: Let $(0, a_1, \ldots, a_{d-1})_n$ denote the function $f_n(x_0, x_1, \ldots, x_{n-1})$ of degree $d$ in $n$ variables generated by the monomial $x_0x_{a_1} \cdots x_{a_{d-1}}$ and having the property that $f_n$ is invariant under cyclic permutations of the variables. Such a function $f_n$ is called monomial rotation symmetric (MRS). Much of this paper extends the work on quadratic MRS functions in a $2020$ paper of the authors to the case of binomial RS functions, that is sums of two quadratic MRS functions. There are also some results for the sum of any number of quadratic MRS functions.

Abstract: A graph $G$ of order $n$ is said to be a sum basic equilibrium if and only if for every edge $uv$ from $G$ and any node $v'$ from $G$, when performing the swap of the edge $uv$ for the edge $uv'$, the sum of the distances from $u$ to all the other nodes is not strictly reduced. This concept lies in the heart of the so-called network creation games, where the central problem is to understand the structure of the resulting equilibrium graphs, and in particular, how well they globally minimize the diameter. It was shown in [Alon, Demaine, Hajiaghayi, Leighton, SIAM J. Discrete Math. 27(2), 2013] that the diameter of sum basic equilibria is $2^{O(\sqrt{\log n})}$ in general, and at most $2$ for trees. In this paper we show that the upper bound of $2$ can be extended to bipartite graphs, and that it also holds for some nonbipartite classes like block graphs and cactus graphs.

Abstract: Let $C_{n_1}\cup C_{n_2}\cup \ldots \cup C_{n_k}$ be a 2-factor i.e. a vertex-disjoint union of cycles. In this note we completely characterize those 2-factors that are uniquely embeddeble in their complement.

Abstract: A weighted graph $\phi G$ encodes a finite metric space $D_{\phi G}$. When is $D$ totally decomposable? When does it embed in $\ell_1$ space? When does its representing matrix have $\leq 1$ positive eigenvalue? We give useful lemmata and prove that these questions can be answered without examining $\phi$ if and only if $G$ has no $K_{2,3}$ minor. We also prove results toward the following conjecture. $D_{\phi G}$ has $\leq n$ positive eigenvalues for all $\phi$, if and only if $G$ has no $K_{2,3,...,3}$ minor, with $n$ threes.

Abstract: We study the inertia of distance matrices of weighted graphs. Our novel congruence-based proof of the inertia of weighted trees extends to a proof for the inertia of weighted unicyclic graphs whose cycle is a triangle. Partial results are given on the inertia of other rationally weighted unicylic graphs.