Yuping Gao — Page 2

arXiv2019

Nonempty intersection of longest paths in graphs without forbidden pairs

In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to Gallai's question for many well-known classes of graphs such as split graphs, series parallel graphs, and $2K_2$-free graphs. Among all the graph classes that support Gallai's question, almost all of them were shown to be Hamiltonian under certain conditions. This observation motivates us to investigate Gallai's question in graphs that are "close" to Hamiltonicity properties. Let $\{R,S\}$ be a pair of connected graphs. In particular, in this paper, we show that Gallai's question is affirmative for all connected $\{R,S\}$-free graphs such that every 2-connected $\{R,S\}$-free graph is Hamiltonian. These pairs $\{R,S\}$ were completely characterized in 1990s.

math.CO↗

arXiv2016

Chromatic index determined by fractional chromatic index

Given a graph $G$ possibly with multiple edges but no loops, denote by $Δ$ the {\it maximum degree}, $μ$ the {\it multiplicity}, $χ'$ the {\it chromatic index} and $χ_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $Δ\le χ_f' \le χ' \le Δ+ μ$, where the upper bound is a classic result of Vizing. While deciding the exact value of $χ'$ is a classic NP-complete problem, the computing of $χ_f'$ is in polynomial time. In fact, it is shown that if $χ_f' > Δ$ then $χ_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H$ of $G$. Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that $χ'=\lceilχ_f'\rceil$ if $χ'\ge Δ+2$, which is commonly referred as Goldberg's conjecture. In this paper, we show that if $χ' >Δ+\sqrt[3]{Δ/2}$ then $χ'=\lceilχ_f'\rceil$. The previous best known result is for graphs with $χ'> Δ+\sqrt{Δ/2}$ obtained by Scheide, and by Chen, Yu and Zang, independently. It has been shown that Goldberg's conjecture is equivalent to the following conjecture of Jakobsen: {\it For any positive integer $m$ with $m\ge 3$, every graph $G$ with $χ'>\frac{m}{m-1}Δ+\frac{m-3}{m-1}$ satisfies $χ'=\lceilχ_f'\rceil$.} Jakobsen's conjecture has been verified for $m$ up to 15 by various researchers in the last four decades. We show that it is true for $m\le 23$. Moreover, we show that Goldberg's conjecture holds for graphs $G$ with $Δ\leq 23$ or $|V(G)|\leq 23$.

math.CO↗