arXiv subjects

Xujun Liu

Publications and source records attributed to Xujun Liu.

25 records · Page 2Linked to original sources

arXiv2019

Packing $(1,1,2,2)$-coloring of some subcubic graphs

For a sequence of non-decreasing positive integers $S = (s_1, \ldots, s_k)$, a packing $S$-coloring is a partition of $V(G)$ into sets $V_1, \ldots, V_k$ such that for each $1\leq i \leq k$ the distance between any two distinct $x,y\in V_i$ is at least $s_i+1$. The smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring is called the packing chromatic number of $G$ and is denoted by $χ_p(G)$. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The question whether $χ_p(D(G)) \le 5$ for all subcubic graphs was first asked by Gastineau and Togni and later conjectured by Bresar, Klavzar, Rall and Wash. Gastineau and Togni observed that if one can prove every subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable then the conjecture holds. The maximum average degree, mad($G$), is defined to be $\max\{\frac{2|E(H)|}{|V(H)|}: H \subset G\}$. In this paper, we prove that subcubic graphs with $mad(G)<\frac{30}{11}$ are packing $(1,1,2,2)$-colorable. As a corollary, the conjecture of Bresar et al holds for every subcubic graph $G$ with $mad(G)<\frac{30}{11}$.

math.CO↗

arXiv2019

Directed Intersection Representations and the Information Content of Digraphs

Consider a directed graph (digraph) in which vertices are assigned color sets, and two vertices are connected if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head. We seek to determine the smallest possible size of the union of the color sets that allows for such a digraph representation. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to introduce the directed intersection number (DIN), the smallest number of colors needed to represent a DAG. Our main results are upper bounds on the DIN of DAGs based on what we call the longest terminal path decomposition of the vertex set, and constructive lower bounds.

math.CO↗

arXiv2019

Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization for large $n$ of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings.

math.CO↗

arXiv2019

Ramsey numbers of ordered graphs under graph operations

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds on the Ramsey number of ordered matchings arising from sum-decomposable permutations, an alternating ordering of the cycle, and an alternating ordering of the tight hyperpath. We also construct ordered matchings on $n$ vertices whose Ramsey number is $n^{q+o(1)}$ for any given exponent $q\in(1,2)$.

math.CO↗

arXiv2019

Cubic graphs with small independence ratio

Let $i(r,g)$ denote the infimum of the ratio $\frac{α(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $α(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \ge 0.4375$, and Csóka improved it to $i(3,\infty) \ge 0.44533$ in 2016. Bollobás proved the upper bound $i(3,\infty) < \frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty) < 0.45537$ in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \le 0.454.$

math.CO↗

arXiv2018

Packing chromatic number of subdivisions of cubic graphs

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The questions on the value of the maximum of $χ_p(G)$ and of $χ_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $χ_p(D(G))\leq 5$ for any subcubic $G$, and later Bresar, Klavzar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $χ_p(G)$ is not bounded in the class of subcubic graphs $G$. In contrast, in this paper we show that $χ_p(D(G))$ is bounded in this class, and does not exceed $8$.

math.CO↗

arXiv2017

Packing chromatic number of subcubic graphs

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ and $g\geq 2k+2$, almost every $n$-vertex cubic graph of girth at least $g$ has the packing chromatic number greater than $k$.

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