Islands in Graphs on Surfaces

Louis Esperet, Pascal Ochem
<span title="">2016</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
An island in a graph is a set X of vertices, such that each element of X has few neighbors outside X. In this paper, we prove several bounds on the size of islands in large graphs embeddable on fixed surfaces. As direct consequences of our results, we obtain that: (1) Every graph of genus g can be colored from lists of size 5, in such a way that each monochromatic component has size O(g). Moreover all but O(g) vertices lie in monochromatic components of size at most 3. (2) Every triangle-free
more &raquo; ... aph of genus g can be colored from lists of size 3, in such a way that each monochromatic component has size O(g). Moreover all but O(g) vertices lie in monochromatic components of size at most 10. (3) Every graph of girth at least 6 and genus g can be colored from lists of size 2, in such a way that each monochromatic component has size O(g). Moreover all but O(g) vertices lie in monochromatic components of size at most 16. While (2) is optimal up to the size of the components, we conjecture that the size of the lists can be decreased to 4 in (1), and the girth can be decreased to 5 in (3). We also study the complexity of minimizing the size of monochromatic components in 2-colorings of planar graphs.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/140957883">doi:10.1137/140957883</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5qnkqk6p7vfu7ocvpenzaprowq">fatcat:5qnkqk6p7vfu7ocvpenzaprowq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191026202809/https://arxiv.org/pdf/1402.2475v2.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b9/b1/b9b1106748873fc3c5bf389ed90e32815405bf36.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/140957883"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>