Interval non-edge-colorable bipartite graphs and multigraphs

An edge-coloring of a graph $G$ with colors $1,...,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In 1991 Erd\H{o}s constructed a bipartite graph with 27 vertices and maximum degree 13 which has no interval coloring. Erd\H{o}s's counterexample is the smallest (in a sense of maximum degree) known bipartite graph which is not interval colorable. On the other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this paper we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs which have no interval coloring, contain 20,19,21 vertices and have maximum degree 11,12,13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.Comment: 18 pages, 7 figure

Identifying and Disentangling Spurious Features in Pretrained Image Representations

Neural networks employ spurious correlations in their predictions, resulting in decreased performance when these correlations do not hold. Recent works suggest fixing pretrained representations and training a classification head that does not use spurious features. We investigate how spurious features are represented in pretrained representations and explore strategies for removing information about spurious features. Considering the Waterbirds dataset and a few pretrained representations, we find that even with full knowledge of spurious features, their removal is not straightforward due to entangled representation. To address this, we propose a linear autoencoder training method to separate the representation into core, spurious, and other features. We propose two effective spurious feature removal approaches that are applied to the encoding and significantly improve classification performance measured by worst group accuracy

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An interval t-coloring of a multigraph G is a proper edge coloring with colors 1, ... , t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cyclic interval t-coloring of a multigraph G is a proper edge coloring with colors 1, ... , t such that the colors of the edges incident with every vertex of G are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. Denote by w(G) (w(c)(G)) and W(G) (W-c(G)) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph G, respectively. We present some new sharp bounds on w(G) and W(G) for multigraphs G satisfying various conditions. In particular, we show that if G is a 2-connected multigraph with an interval coloring, then W(G) amp;lt;= 1 + left perpendicular vertical bar V(G)vertical bar/2 right perpendicular (Delta(G) - 1). We also give several results towards the general conjecture that W-c(G) amp;lt;= I vertical bar V(G)vertical bar for any triangle-free graph G with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most 4. (C) 2017 Elsevier B.V. All rights reserved

Interval Edge-Colorings of Cartesian Products of Graphs I

A proper edge-coloring of a graph $G$ with colors $1, . . ., t$ is an interval $t$-coloring if all colors are used and the colors of edges incident to each vertex of $G$ form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G \in \mathfrak{N}$, the least and the greatest values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$, respectively. In this paper we first show that if $G$ is an $r$-regular graph and $G \in \mathfrak{N}$, then $W(G \square P_m) \geq W(G) + W(P_m) + (m − 1)r$ $(m \in \mathbb{N})$ and $W(G \square C_{2n}) \geq W(G) +W(C_{2n}) + nr$ $(n \geq 2)$. Next, we investigate interval edge-colorings of grids, cylinders and tori. In particular, we prove that if $G \square H$ is planar and both factors have at least 3 vertices, then $G \square H \in \mathfrak{N}$ and $w(G \square H) leq 6$. Finally, we confirm the first author’s conjecture on the $n$-dimensional cube $Q_n$ and show that $Q_n$ has an interval $t$-coloring if and only if $n \leq t \leq \frac{n(n+1)}{2}$