arXiv subjects
Publications and source records attributed to Gexin Yu.
An open-locating-dominating set (OLD-set) is a subset of vertices of a graph such that every vertex in the graph has at least one neighbor in the set and no two vertices in the graph have the same set of neighbors in the set. This is an analogue to the well-studied identifying code in the literature. In this paper, we prove that the optimal density of the OLD-set for the infinite triangular grid is $4/13$.
We consider the problem of upper bounding the number of circular transpositions needed to sort a permutation. It is well known that any permutation can be sorted using at most $n(n-1)/2$ adjacent transpositions. We show that, if we allow all adjacent transpositions, as well as the transposition that interchanges the element in position 1 with the element in the last position, then the number of transpositions needed is at most $n^2/4$. This answers an open question posed by Feng, Chitturi and Sudborough (2010).
We prove that the strong chromatic index for each $k$-degenerate graph with maximum degree $Δ$ is at most $(4k-2)Δ-k(2k-1)+1$.
A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring.
A graph has a perfect partition if all its perfect matchings can be partitioned so that each part is a 1-factorization of the graph. Let $L_{rm, r}=K_{rm,rm}-mK_{r,r}$. We first give a formula to count the number of perfect matchings of $L_{rm, r}$, then show that $L_{6,1}$ and $L_{8,2}$ have perfect partitions.
A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar graph without 4- and 5-cycles is $(1, 1, 0)$-colorable and $(3,0,0)$-colorable. This is a relaxation of the Steinberg Conjecture that every planar graph without 4- and 5-cycles are properly 3-colorable (i.e., $(0,0,0)$-colorable).
For a graph, $G$, and a vertex $v \in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \in V(G)$, the vertex sets $N[v_1] \cap D$ and $N[v_2] \cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \approx 0.413793$ and was proved by Cranston and Yu in 2009. We present three new codes with a density of 3/7, and we improve the lower bound to $5/12 \approx 0.416667$.
We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $Δ(G)$ satisfies $\lcl(G)\ge \ceil{Δ(G)/2}+1$. In this paper, we prove the following results: (1) if $\mad(G)<12/5$ and $Δ(G)\ge 3$, then $\lcl(G)=\ceil{Δ(G)/2}+1$, and we give an infinite family of examples to show that this result is best possible; (2) if $\mad(G)<3$ and $Δ(G)\ge 9$, then $\lcl(G)\le\ceil{Δ(G)/2}+2$, and we give an infinite family of examples to show that the bound on $\mad(G)$ cannot be increased in general; (3) if $G$ is planar and has girth at least 5, then $\lcl(G)\le\ceil{Δ(G)/2}+4$.
Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $χ_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $χ_i(G)\leqΔ(G) + 1$; and if $mad(G) < 42/19$, then $χ_i(G)=Δ(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $Δ(G)\geq 4$. We prove that if $g(G)\geq 9$, then $χ_i(G)\leqΔ(G)+1$; similarly, if $g(G)\geq 13$, then $χ_i(G)=Δ(G)$.
Let $\mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $χ_i(G)$ denote the injective chromatic number of $G$. We prove that if $Δ\geq 4$ and $\mad(G)<\frac{14}5$, then $χ_i(G)\leqΔ+2$. When $Δ=3$, we show that $\mad(G)<\frac{36}{13}$ implies $χ_i(G)\le 5$. In contrast, we give a graph $G$ with $Δ=3$, $\mad(G)=\frac{36}{13}$, and $χ_i(G)=6$.
Given a graph $G$, an identifying code $C \subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap C$ and $N[v_2]\cap C$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\approx0.413793$.