arXiv subjects

Maria Axenovich

Publications and source records attributed to Maria Axenovich.

76 records · Page 5Linked to original sources

arXiv2012

On homometric sets in graphs

For a vertex set $S\subseteq V(G)$ in a graph $G$, the {\em distance multiset}, $D(S)$, is the multiset of pairwise distances between vertices of $S$ in $G$. Two vertex sets are called {\em homometric} if their distance multisets are identical. For a graph $G$, the largest integer $h$, such that there are two disjoint homometric sets of order $h$ in $G$, is denoted by $h(G)$. We slightly improve the general bound on this parameter introduced by Albertson, Pach and Young (2010) and investigate it in more detail for trees and graphs of bounded diameter. In particular, we show that for any tree $T$ on $n$ vertices $h(T) \geq \sqrt[3]{n}$ and for any graph $G$ of fixed diameter $d$, $h(G) \geq cn^{1/ (2d-2)}$.

math.CO↗

arXiv2012

Fork-forests in bi-colored complete bipartite graphs

Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2-edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of $G=K_{n,n}$ are colored with black and white such that the number of black edges differs from the number of white edges by at most 1, then there are at least $n(1-1/\sqrt{2})$ vertex-disjoint forks with centers in the same partite set of $G$. Here, a fork is a graph formed by two adjacent edges of different colors. The bound is sharp. Moreover, an algorithm running in time $O(n^2 \log n \sqrt{n α(n^2,n) \log n})$ and giving a largest such fork forest is found.

cs.DM↗

arXiv2011

List precoloring extension in planar graphs

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thomassen's theorem can be extended by precoloring some vertices which are at a large enough distance apart in a graph. Here, among others, we answer the question in the case when the graph does not contain short cycles separating precolored vertices and when there is a "wide" Steiner tree containing all the precolored vertices.

math.CO↗

arXiv2010

A note on monotonicity of mixed Ramsey numbers

For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some $(G,H)$-colorings of $K_n$ is called a mixed-Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answer is "yes" if $G$ is not a star and $H$ does not contain a pendent edge.

math.CO↗