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Hung-Lin Fu

Publications and source records attributed to Hung-Lin Fu.

17 recordsLinked to original sources
arXiv2022

Epidemic Source Detection in Contact Tracing Networks: Epidemic Centrality in Graphs and Message-Passing Algorithms

We study the epidemic source detection problem in contact tracing networks modeled as a graph-constrained maximum likelihood estimation problem using the susceptible-infected model in epidemiology. Based on a snapshot observation of the infection subgraph, we first study finite degree regular graphs and regular graphs with cycles separately, thereby establishing a mathematical equivalence in maximal likelihood ratio between the case of finite acyclic graphs and that of cyclic graphs. In particular, we show that the optimal solution of the maximum likelihood estimator can be refined to distances on graphs based on a novel statistical distance centrality that captures the optimality of the nonconvex problem. An efficient contact tracing algorithm is then proposed to solve the general case of finite degree-regular graphs with multiple cycles. Our performance evaluation on a variety of graphs shows that our algorithms outperform the existing state-of-the-art heuristics using contact tracing data from the SARS-CoV 2003 and COVID-19 pandemics by correctly identifying the superspreaders on some of the largest superspreading infection clusters in Singapore and Taiwan.

cs.SI↗
arXiv2021

Total Colorings of Some Classes of FourRegular Circulant Graphs

The total chromatic number, $\chi''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph, $\chi''(G) \leq \Delta(G)+2 $, where $\Delta(G)$ represents the maximum degree of $G$. In this paper we obtained the total chromatic number for some classes of four regular circulant graphs.

math.CO↗
arXiv2020

Strongly separable matrices for nonadaptive combinatorial group testing

In nonadaptive combinatorial group testing (CGT), it is desirable to identify a small set of up to $d$ defectives from a large population of $n$ items with as few tests (i.e. large rate) and efficient identifying algorithm as possible. In the literature, $d$-disjunct matrices ($d$-DM) and $\bar{d}$-separable matrices ($\bar{d}$-SM) are two classical combinatorial structures having been studied for several decades. It is well-known that a $d$-DM provides a more efficient identifying algorithm than a $\bar{d}$-SM, while a $\bar{d}$-SM could have a larger rate than a $d$-DM. In order to combine the advantages of these two structures, in this paper, we introduce a new notion of \emph{strongly $d$-separable matrix} ($d$-SSM) for nonadaptive CGT and show that a $d$-SSM has the same identifying ability as a $d$-DM, but much weaker requirements than a $d$-DM. Accordingly, the general bounds on the largest rate of a $d$-SSM are established. Moreover, by the random coding method with expurgation, we derive an improved lower bound on the largest rate of a $2$-SSM which is much higher than the best known result of a $2$-DM.

math.CO↗
arXiv2020

Beyond Hamiltonicity of Prime Difference Graphs

A graph is Hamiltonian if it contains a cycle which visits every vertex of the graph exactly once. In this paper, we consider the problem of Hamiltonicity of a graph $G_n$, which will be called the prime difference graph of order $n$, with vertex set $\{1,2,\cdots, n\}$ and edge set $\{uv: |u-v|$ is a prime number$\}$. A recent result, conjectured by Sun and later proved by Chen, asserts that $G_n$ is Hamiltonian for $n\geq 5$. This paper extends their result in three directions. First, we prove that for any two integers $a$ and $b$ with $1\leq a<b\leq n$, there is a Hamilton path in $G_n$ from $a$ to $b$ except some cases of small $n$. This result implies robustness of the Hamiltonicity property of the prime difference graph in a sense that for any edge $e$ in $G_n$ there exists a Hamilton cycle containing $e$. Second, we show that the prime difference graph contains considerably more about the cycle structure than Hamiltonicity; precisely, for any integer $n\geq 7$, the prime difference graph $G_n$ contains any 2-factor of the complete graph of order $n$ as a subgraph. Finally, we find that $G_n$ may contain more edge-disjoint Hamilton cycles. In particular, these Hamilton cycles are generated by two prime differences.

math.CO↗
arXiv2019

On the integer {k}-domination number of circulant graphs

Let $G=(V,E)$ be a simple undirected graph. $G$ is a circulant graph defined on $V=\mathbb{Z}_n$ with difference set $D\subseteq \{1,2,\ldots,\lfloor\frac{n}{2}\rfloor\}$ provided two vertices $i$ and $j$ in $\mathbb{Z}_n$ are adjacent if and only if $\min\{|i-j|, n-|i-j|\}\in D$. For convenience, we use $G(n;D)$ to denote such a circulant graph. A function $f:V(G)\rightarrow\mathbb{N}\cup\{0\}$ is an integer $\{k\}$-domination function if for each $v\in V(G)$, $\sum_{u\in N_G[v]}f(u)\geq k.$ By considering all $\{k\}$-domination functions $f$, the minimum value of $\sum_{v\in V(G)}f(v)$ is the $\{k\}$-domination number of $G$, denoted by $\gamma_k(G)$. In this paper, we prove that if $D=\{1,2,\ldots,t\}$, $1\leq t\leq \frac{n-1}{2}$, then the integer $\{k\}$-domination number of $G(n;D)$ is $\lceil\frac{kn}{2t+1}\rceil$.

math.CO↗
arXiv2018

The Undirected Optical Indices of Trees

For a connected graph $G$, an instance $I$ is a set of pairs of vertices and a corresponding routing $R$ is a set of paths specified for all vertex-pairs in $I$. Let $\mathfrak{R}_I$ be the collection of all routings with respect to $I$. The undirected optical index of $G$ with respect to $I$ refers to the minimum integer $k$ to guarantee the existence of a mapping $\phi:R\to\{1,2,\ldots,k\}$, such that $\phi(P)\neq\phi(P')$ if $P$ and $P'$ have common edge(s), over all routings $R\in\mathfrak{R}_I$. A natural lower bound of the undirected optical index is the edge-forwarding index, which is defined to be the minimum of the maximum edge-load over all possible routings. Let $w(G,I)$ and $\pi(G,I)$ denote the undirected optical index and edge-forwarding index with respect to $I$, respectively. In this paper, we derive the inequality $w(T,I_A)<\frac{3}{2}\pi(T,I_A)$ for any tree $T$, where $I_A:=\{\{x,y\}:\,x,y\in V(T)\}$ is the all-to-all instance.

math.CO↗
arXiv2018

From a Consequence of Bertrand's Postulate to Hamilton Cycles

A consequence of Bertrand's postulate, proved by L. Greenfield and S. Greenfield in 1998, assures that the set of integers $\{1,2,\cdots, 2n\}$ can be partitioned into pairs so that the sum of each pair is a prime number for any positive integer $n$. Cutting through it from the angle of Graph Theory, this paper provides new insights into the problem. We conjecture a stronger statement that the set of integers $\{1,2,\cdots, 2n\}$ can be rearranged into a cycle so that the sum of any two adjacent integers is a prime number. Our main result is that this conjecture is true for infinitely many cases.

math.CO↗
arXiv2017

The Undirected Optical Indices of Complete $m$-ary Trees

The routing and wavelength assignment problem arises from the investigation of optimal wavelength allocation in an optical network that employs Wavelength Division Multiplexing (WDM). Consider an optical network that is represented by a connected, simple graph $G$. An all-to-all routing $R$ in $G$ is a set of paths connecting all pairs of vertices of $G$. The undirected optical index of $G$ is the minimum integer $k$ to guarantee the existence of a mapping $\phi:R\to\{1,2,\ldots,k\}$, such that $\phi(P)\neq\phi(P')$ if $P$ and $P'$ have common edge(s), over all possible routings $R$. A natural lower bound of the undirected optical index of $G$ is the (undirected) edge-forwarding index, which is defined to be the minimum of the maximum edge-load over all possible all-to-all routings. In this paper, we first derive the exact value of the optical index of the complete $m$-ary trees, and then investigate the gap between undirected optical and edge-forwarding indices.

math.CO↗
arXiv2016

On the Number of Rainbow Spanning Trees in Edge-Colored Complete Graphs

A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if $K_{2m}$ is properly $(2m-1)$-edge-colored, then the edges of $K_{2m}$ can be partitioned into $m$ rainbow spanning trees except when $m=2$. By means of an explicit, constructive approach, in this paper we construct $\lfloor \sqrt{6m+9}/3 \rfloor$ mutually edge-disjoint rainbow spanning trees for any positive value of $m$. Not only are the rainbow trees produced, but also some structure of each rainbow spanning tree is determined in the process. This improves upon best constructive result to date in the literature which produces exactly three rainbow trees.

math.CO↗
arXiv2015

The Global Packing Number for an Optical Network

The global packing number problem arises from the investigation of optimal wavelength allocation in an optical network that employs Wavelength Division Multiplexing (WDM). Consider an optical network that is represented by a connected, simple graph $G$. We assume all communication channels are bidirectional, so that all links and paths are undirected. It follows that there are ${|G|\choose 2}$ distinct node pairs associated with $G$, where $|G|$ is the number of nodes in $G$. A path system $\mathcal{P}$ of $G$ consists of ${|G|\choose 2}$ paths, one path to connect each of the node pairs. The global packing number of a path system $\mathcal{P}$, denoted by $\Phi(G,\mathcal{P})$, is the minimum integer $k$ to guarantee the existence of a mapping $\omega:\mathcal{P}\to\{1,2,\ldots,k\}$, such that $\omega(P)\neq\omega(P')$ if $P$ and $P'$ have common edge(s). The global packing number of $G$, denoted by $\Phi(G)$, is defined to be the minimum $\Phi(G,\mathcal{P})$ among all possible path systems $\mathcal{P}$. If there is no wavelength conversion along any optical transmission path for any node pair in the network, the global packing number signifies the minimum number of wavelengths required to support simultaneous communication for all pairs in the network. In this paper, the focus is on ring networks, so that $G$ is a cycle. Explicit formulas for the global packing number of a cycle is derived. The investigation is further extended to chain networks. A path system, $\mathcal{P}$, that enjoys $\Phi(G,\mathcal{P})=\Phi(G)$ is called ideal. A characterization of ideal path systems is also presented. We also describe an efficient heuristic algorithm to assign wavelengths that can be applied to a general network with more complicated traffic load.

cs.DM↗
arXiv2015

The Hamilton-Waterloo Problem for Triangle-Factors and Heptagon-Factors

Given 2-factors $R$ and $S$ of order $n$, let $r$ and $s$ be nonnegative integers with $r+s=\lfloor \frac{n-1}{2}\rfloor$, the Hamilton-Waterloo problem asks for a 2-factorization of $K_n$ if $n$ is odd, or of $K_n-I$ if $n$ is even, in which $r$ of its 2-factors are isomorphic to $R$ and the other $s$ 2-factors are isomorphic to $S$. In this paper, we solve the problem for the case of triangle-factors and heptagon-factors for odd $n$ with 3 possible exceptions when $n=21$.

math.CO↗
arXiv2014

Codes with the Identifiable Parent Property for Multimedia Fingerprinting

Let ${\cal C}$ be a $q$-ary code of length $n$ and size $M$, and ${\cal C}(i) = \{{\bf c}(i) \ | \ {\bf c}=({\bf c}(1), {\bf c}(2), \ldots, {\bf c}(n))^{T} \in {\cal C}\}$ be the set of $i$th coordinates of ${\cal C}$. The descendant code of a sub-code ${\cal C}^{'} \subseteq {\cal C}$ is defined to be ${\cal C}^{'}(1) \times {\cal C}^{'}(2) \times \cdots \times {\cal C}^{'}(n)$. In this paper, we introduce a multimedia analogue of codes with the identifiable parent property (IPP), called multimedia IPP codes or $t$-MIPPC$(n, M, q)$, so that given the descendant code of any sub-code ${\cal C}^{'}$ of a multimedia $t$-IPP code ${\cal C}$, one can always identify, as IPP codes do in the generic digital scenario, at least one codeword in ${\cal C}^{'}$. We first derive a general upper bound on the size $M$ of a multimedia $t$-IPP code, and then investigate multimedia $3$-IPP codes in more detail. We characterize a multimedia $3$-IPP code of length $2$ in terms of a bipartite graph and a generalized packing, respectively. By means of these combinatorial characterizations, we further derive a tight upper bound on the size of a multimedia $3$-IPP code of length $2$, and construct several infinite families of (asymptotically) optimal multimedia $3$-IPP codes of length $2$.

cs.IT↗
arXiv2014

Partially user-irrepressible sequence sets and conflict-avoiding codes

In this paper we give a partial shift version of user-irrepressible sequence sets and conflict-avoiding codes. By means of disjoint difference sets, we obtain an infinite number of such user-irrepressible sequence sets whose lengths are shorter than known results in general. Subsequently, the newly defined partially conflict-avoiding codes are discussed.

cs.IT↗
arXiv2014

Multicolored Isomorphic Spanning Trees in Complete Graphs

In this paper, we first prove that if the edges of $K_{2m}$ are properly colored by $2m-1$ colors in such a way that any two colors induce a 2-factor of which each component is a 4-cycle, then $K_{2m}$ can be decomposed into $m$ isomorphic multicolored spanning trees. Consequently, we show that there exist three disjoint isomorphic multicolored spanning trees in any properly (2$m-$1)-edge-colored $K_{2m}$ for $m\geq 14$.

math.CO↗
arXiv2014

Edge-colorings of $K_{m,n}$ which Forbid Multicolored Cycles

A subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this paper, we study the proper edge-colorings of the complete bipartite graph $K_{m,n}$ which forbid multicolored cycles. Mainly, we prove that (1) for any integer $k\geq 2$, if $n\geq 5k-6$, then any properly $n$-edge-colored $K_{k,n}$ contains a multicolored $C_{2k}$, and (2) determine the order of the properly edge-colored complete bipartite graphs which forbid multicolored $C_6$.

math.CO↗
arXiv2012

Sparseness of 4-cycle systems

An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erd\H{o}s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it contains no configurations consisting of j 4-cycles whose union contains precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a configuration on two 4-cycles sharing a diagonal, called the double-diamond, we say it is strictly r-sparse. In this paper, we show that for every admissible order v there exists a strictly 4-sparse 4CS(v). We also prove that for any positive integer r > 1 and sufficiently large integer v there exists a constant number c such that there exists a strictly r-sparse 4-cycle packing of order v with cv^2 4-cycles.

math.CO↗