Short Cycle Covers of Graphs with at Most 77% Vertices of Degree Two

Let $G$ be a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two and let $cc(G)$ be the length of its shortest cycle cover. It is known that if $cc(G) < 1.4m$ in bridgeless graphs with $n_2 \le m/10$, then the Cycle Double Cover Conjecture holds. Fan (2017) proved that if $n_2 = 0$, then $cc(G) < 1.6258m$ and $cc(G) < 1.6148m$ provided that $G$ is loopless; morever, if $n_2 \le m/30$, then $cc(G) < 1.6467m$. We show that for a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two, $cc(G) < 1.6148m + 0.0741n_2$. Therefore, if $n_2=0$, then $cc(G) < 1.6148m$ even if $G$ has loops; if $n_2 \le m/30$, then $cc(G) < 1.6173m$; and if $n_2 \le m/10$, then $cc(G) < 1.6223|E(G)|$. Our improvement is obtained by randomizing Fan's construction.

Idealness of k-wise intersecting families

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer $$k\ge 4$$ k ≥ 4 , every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $$k=4$$ k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters.

Meningkatkan Keaktifan dan Hasil Belajar Materi Himpunan melalui Model Pembelajaran Jigsaw Berbantuan Android

This study aims to improve the activeness as well as mathematics learning achievement on sets topic through the Jigsaw learning model, grade VII B students of SMP Negeri 12 Semarang odd semester of 2018/2019 academic year. The research process consists of two cycles, in which each cycle covers for steps namely planning, action, observation and reflection. The research data were in the form of observation sheet notes, test results, and assignment results. The type of collected data is qualitative and quantitative data. Technique of data collection applies documentation, observation and analysis of learning results. Research instruments cover score list, notes of teaching and learning journal, and observation sheet. Data validation is done by involving observer and colleagues involved during the collaboration. Data analysis was performed using descriptive comparative technique by comparing pre, first and second cycle data. This step was then followed with critical analysis by conducting the reflection. Previously, the average knowledge score of grade VII B students on set topic was 55, while the average skill score was 56. Later on, after the jigsaw learning model was applied, average of the knowledge score increased to 65.6 at the first cycle, and 81.3 at the second cycle. Meanwhile the average of skill score was increased to 71.9 at the first cycle and 83.6 at the second cycle. These results indicate that there is an increase at the mathematics learning achievement for grade VII B students. In addition, students’ activities are increasingly optimal; they show quite positive responses towards teaching and learning activity.