arXiv subjects

Hoang La

Publications and source records attributed to Hoang La.

23 records · Page 2Linked to original sources

arXiv2021

$2$-distance list $(Δ+2)$-coloring of planar graphs with girth at least 10

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say that a graph is list $k$-colorable when every vertex $v$ has a list of colors of size at least $k$. A $2$-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance list ($Δ+2$)-coloring for planar graphs with girth at least $10$ and maximum degree $Δ\geq 4$.

math.CO↗

arXiv2021

$2$-distance $(Δ+2)$-coloring of sparse graphs

A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($Δ+2$)-coloring for graphs with maximum average degree less than $\frac{8}{3}$ (resp. $\frac{14}{5}$) and maximum degree $Δ\geq 6$ (resp. $Δ\geq 10$). As a corollary, every planar graph with girth at least $8$ (resp. $7$) and maximum degree $Δ\geq 6$ (resp. $Δ\geq 10$) admits a $2$-distance $(Δ+2)$-coloring.

math.CO↗

arXiv2021

A note on deterministic zombies

"Zombies and Survivor" is a variant of the well-studied game of "Cops and Robber" where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path if multiple options are available. The zombie number, like the cop number, of a graph is the minimum number of zombies, or cops, required to capture the survivor. In this short note, we solve a question by Fitzpatrick et al., proving that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers. We also give a simple graph family with cop number $2$ and an arbitrarily large zombie number.

math.CO↗

arXiv2021

2-distance list $(Δ+ 3)$-coloring of sparse graphs

A 2-distance list k-coloring of a graph is a proper coloring of the vertices where each vertex has a list of at least k available colors and vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance list $(Δ+ 3)$-coloring for graphs with maximum average degree less than $\frac83$ and maximum degree $Δ\geq 4$ as well as graphs with maximum average degree less than $\frac{14}5$ and maximum degree $Δ\geq 6$.

math.CO↗

arXiv2021

$2$-distance $(Δ+1)$-coloring of sparse graphs using the potential method

A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($Δ+1$)-coloring for graphs with maximum average degree less than $\frac{18}{7}$ and maximum degree $Δ\geq 7$. As a corollary, every planar graph with girth at least $9$ and $Δ\geq 7$ admits a $2$-distance $(Δ+1)$-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on $2$-distance coloring.

math.CO↗