Local Crossing Numbers of the Product of Planar Graphs and Cycles

Download or read book Local Crossing Numbers of the Product of Planar Graphs and Cycles written by Marine Musulyan and published by . This book was released on 2019 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graph is said to be planar if it can be drawn in the plane so that its edges intersect only at their ends. The crossing number of a graph is the minimum number of edge-crossings over all its drawings. The local crossing number of a graph is the minimum value of k such that there is a drawing of the graph in which all its edges are crossed at most k times. While there have been several developments on the crossing number for products of graphs, virtually nothing is known about their local crossing number. We prove several results about the local crossing number, lcr(GxH), of the product of two graphs G and H. In particular, when G is a planar graph, like the star Sn, and H is a path Pn or a cycle Cn. We prove that lcr(CmxCn) =1 and complete the list of exact values of lcr(GxCn) where G is any graph with at most 4 vertices and 5 edges. Our main work investigates lcr(SmxCn) and lcr(SmxPn). In regards to cycles, we prove that 2 ≤ lcr(Sm x Cn) ≤ m/2-1 for m>7 and n>5. We conjecture that lcr(Sm x Cn)=m/2-1 and prove this conjecture when m≤ 7 and n≥4. In regards to paths, we prove 1 ≤ lcr(Sm x Pn) ≤ m/2-1 for m≥4 and n≥3. Finally, we find a non-trivial drawings of the product S6 x P4 with local crossing number 1, which in turn proves that lcr(S6 x P4)=1.