报告题目：Homomorphism bound of partial t-trees

报告简介：A reformulation of the four color theorem is to say that $K_4$ is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, Naserasr has proved (using the four color theorem) that the Clebsch graph is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension $2k$, $PC(2k)$, is a smallest graph of odd-girth $2k+1$ to which every planar graph of odd-girth at least $2k+1$ admits a homomorphism. This conjecture is believed to be true for the larger class of $K_5$-minor-free graphs (which includes the class of planar graphs). Motivated by this conjecture and in extension of a recent work of L. Beaudou, F. Foucaud and R. Naserasr, which studies homomorphism bounds for the class of $K_4$-minor-free graphs, in this talk we present a necessary and sufficient condition for a graph $B$ of odd-girth $2k+1$ to admit a homomorphism from any partial $t$-tree of odd-girth at least $2k+1$. Applying our results to the class of partial 3-trees, which is a rich subclass of $K_5$-minor-free graphs, we show that $PC(2k)$ is in fact a smallest graph of odd-girth $2k+1$ to which every partial 3-tree of odd-girth at least $2k+1$ admits a homomorphism. We then apply this result to show that every planar $(2k+1)$-regular multigraph $G$ whose dual is a partial 3-tree, and whose fractional edge-chromatic number is $2k+1$, is $(2k+1)$-edge-colorable.

报告人：陈美润，2009年博士毕业于厦门大学，现任职厦门理工学院；读博和工作期间多次访问巴黎sacley大学(原来的巴黎十一大学)和巴黎大学(原来的巴黎七大)，研究领域: 图染色，网络的诊断度。

报告时间：2021年1月27日（星期三）下午 2:00-3:30.

报告地点：腾讯会议，ID：733 669 976

主办单位：扬州大学数学科学学院

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