arXiv subjects

Shinya Fujita

Publications and source records attributed to Shinya Fujita.

At least 19 recordsLinked to original sources

arXiv2026

From Halin's Edge Removability to Matching Removability in $k$-Connected Graphs

We study matching-removability under the degree/connectivity regime of Halin's theorem, which asserts that every $k$-connected graph $G$ with minimum degree $\delta(G)\ge k+1$ contains an edge $e$ such that $G-e$ remains $k$-connected. For $k,\ell\ge 1$, an $\ell$-matching is a matching of size $\ell$. A matching $M$ in a $k$-connected graph $G$ is {\it $k$-removable} if $G-M$ remains $k$-connected. We improve Halin's result by proving that every $k$-connected graph $G$ with $\delta(G)\ge k+1$ contains a $k$-removable $2$-matching, except when $k=1$ and $G$ is a cycle. For small $k$ we obtain stronger bounds: (i) $k=1$: a 1-removable $\min\{\lfloor n/2\rfloor,\delta(G)\}$-matching; (ii) $k=2$: a 2-removable $\lceil(\delta(G)+1)/2\rceil$-matching, with a unique tight exception when $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$; and (iii) $k=3$: for $\delta(G)\ge 5$, a $3$-removable $\lceil(\delta(G)+1)/2\rceil$-matching. All these bounds are optimal with respect to removable matching size and minimum degree. We also show that for every $n\ge 2\delta$, there exists a $k$-connected $n$-vertex graph $G$ with minimum degree $\delta$ that does not contain a $k$-removable matching of size at least $\delta(G)+1$. Moreover, for $k\le 2$ there exists a $k$-removable $(\delta(G)-c)$-matching for some $c\le 3$, which is optimal up to the additive constant.

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arXiv2026

Lessons Learned from Structural Design and Vibration Testing of 50-kg Microsatellites Deployed from the International Space Station

Hokkaido University and Tohoku University have been developing and operating a constellation of 50-cm-class microsatellites for Earth observation. DIWATA-1, launched in 2016, was deployed into a circular orbit at an altitude of approximately 400 km from the International Space Station (ISS). For the subsequent satellite developed in 2021, the structural design and vibration test campaign were optimized to meet a strict one-year development schedule. This paper summarizes how the structural design of the previous satellite was reviewed and updated, and how the vibration test was successfully completed in a single trial to minimize schedule and technical risks. These lessons learned provide valuable insights, as there are only a limited number of reported cases of 50-kg-class microsatellites deployed from the ISS.

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arXiv2026

System Analysis and Pre-Flight Evaluation of Deployable Solar Panels for 3U CubeSat HOKUSHIN-1

This paper describes the system design methodology derived from the development and evaluation tests of deployable solar panels to be mounted on a 3U CubeSat. The study mainly includes structural analysis, thermal analysis, and a review of vibration test results. Hokkaido University is developing the 3U CubeSat HOKUSHIN-1 in collaboration with Tohoku University and Muroran Institute of Technology. Deployable solar panels are a key technology for future planned lunar exploration missions, as they enable power-intensive communication and propulsion required for orbit control. The satellite also demonstrates a newly developed compact and efficient propulsion system. The satellite has dimensions of approximately 10x10x34 cm, a mass of 3.99 kg, and will be deployed into a circular orbit at an altitude of about 400 km with an orbital inclination of 51.6 degrees from the International Space Station.

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arXiv2025

Connectivity keeping trees in triangle-free graphs

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022, Luo, Tian, and Wu considered an analogous problem for bipartite graphs and conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+\max\{|X|,|Y|\}$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In this paper, we relax the bipartite assumption by considering triangle-free graphs and prove that for any tree $T$ of order $m$, every $k$-connected triangle-free graph $G$ with minimum degree at least $2k+3m-4$ contains a subtree $T' \cong T$ such that $G-V(T')$ remains $k$-connected. Furthermore, we establish refined results for specific subclasses such as bipartite graphs or graphs with girth at least five.

math.CO↗

arXiv2025

Tight upper bounds on the hop domination number of triangle-free graphs

For a graph $G$, a subset $S$ of $V(G)$ is a {\it hop dominating set} of $G$ if every vertex not in $S$ has a $2$-step neighbor in $S$. The {\it hop domination number}, $γ_h(G)$, of $G$ is the minimum cardinality of a hop dominating set of $G$. In this paper, we show that for a connected triangle-free graph $G$ with $n\ge 15$ vertices, if $δ(G)\ge 2$, then $γ_h(G)\le \frac{2n}{5}$, and the bound is tight. We also give some tight upper bounds on $γ_h(G)$ for {triangle-free} graphs $G$ that contain a Hamiltonian path or a Hamiltonian cycle.

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arXiv2024

Monochromatic $k$-connection of graphs

An edge-coloured path is monochromatic if all of its edges have the same colour. For a $k$-connected graph $G$, the monochromatic $k$-connection number of $G$, denoted by $mc_k(G)$, is the maximum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by $k$ internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter $mc_k(G)$. We obtain bounds for $mc_k(G)$, for general graphs $G$. We also compute $mc_k(G)$ exactly when $k$ is small, and $G$ is a graph on $n$ vertices, with a spanning $k$-connected subgraph having the minimum possible number of edges, namely $\lceil\frac{kn}{2}\rceil$. We prove a similar result when $G$ is a bipartite graph.

math.CO↗

arXiv2022

New classification of graphs in view of the domination number of central graphs

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $γ(G)$ and $τ(G)$ be the domination number of $G$ and the minimum cardinarity of a vertex cover of $G$, respectively. In this paper, we give a new classification of graphs concerning the domination number of central graphs and minimum vertex covers of graphs. Namely, we show that any graph $G$ with at least three vertices can be classified into one of the two classes of graphs with $γ(C(G))=τ(G)$ and $γ(C(G))=τ(G)+1$, respectively, together with some special properties concerning a vertex cover of $G$. We also give some new results on the domination number of central graphs.

math.CO↗

arXiv2021

On properly ordered coloring of vertices in a vertex-weighted graph

We introduce the notion of a properly ordered coloring (POC) of a weighted graph, that generalizes the notion of vertex coloring of a graph. Under a POC, if $xy$ is an edge, then the larger weighted vertex receives a larger color; in the case of equal weights of $x$ and $y$, their colors must be different. In this paper, we shall initiate the study of this special coloring in graphs. For a graph $G$, we introduce the function $f(G)$ which gives the maximum number of colors required by a POC over all weightings of $G$. We show that $f(G)=\ell(G)$, where $\ell(G)$ is the number of vertices of a longest path in $G$. Another function we introduce is $χ_{POC}(G;t)$ giving the minimum number of colors required over all weightings of $G$ using $t$ distinct weights. We show that the ratio of $χ_{POC}(G;t)-1$ to $χ(G)-1$ can be bounded by $t$ for any graph $G$; in fact, the result is shown by determining $χ_{POC}(G;t)$ when $G$ is a complete multipartite graph. We also determine the minimum number of colors to give a POC on a vertex-weighted graph in terms of the number of vertices of a longest directed path in an orientation of the underlying graph. This extends the so called Gallai-Hasse-Roy-Vitaver theorem, a classical result concerning the relationship between the chromatic number of a graph $G$ and the number of vertices of a longest directed path in an orientation of $G$.

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arXiv2020

The size of graphs with restricted rainbow $2$-connection number

Let $k$ be a positive integer, and $G$ be a $k$-connected graph. An edge-coloured path is \emph{rainbow} if all of its edges have distinct colours. The \emph{rainbow $k$-connection number} of $G$, denoted by $rc_k(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by $k$ internally vertex-disjoint rainbow paths. The function $rc_k(G)$ was introduced by Chartrand, Johns, McKeon and Zhang in 2009, and has since attracted significant interest. Let $t_k(n,r)$ denote the minimum number of edges in a $k$-connected graph $G$ on $n$ vertices with $rc_k(G)\le r$. Let $s_k(n,r)$ denote the maximum number of edges in a $k$-connected graph $G$ on $n$ vertices with $rc_k(G)\ge r$. The functions $t_1(n,r)$ and $s_1(n,r)$ have previously been studied by various authors. In this paper, we study the functions $t_2(n,r)$ and $s_2(n,r)$. We determine bounds for $t_2(n,r)$ which imply that $t_2(n,2)=(1+o(1))n\log_2 n$, and $t_2(n,r)$ is linear in $n$ for $r\ge 3$. We also provide some remarks about the function $s_2(n,r)$.

math.CO↗

arXiv2020

The optimal proper connection number of a graph with given independence number

An edge-colored connected graph $G$ is properly connected if between every pair of distinct vertices, there exists a path that no two adjacent edges have a same color. Fujita (2019) introduced the optimal proper connection number ${\mathrm{pc}_{\mathrm{opt}}}(G)$ for a monochromatic connected graph $G$, to make a connected graph properly connected efficiently. More precisely, ${\mathrm{pc}_{\mathrm{opt}}}(G)$ is the smallest integer $p+q$ when one converts a given monochromatic graph $G$ into a properly connected graph by recoloring $p$ edges with $q$ colors. In this paper, we show that ${\mathrm{pc}_{\mathrm{opt}}}(G)$ has an upper bound in terms of the independence number $α(G)$. Namely, we prove that for a connected graph $G$, ${\mathrm{pc}_{\mathrm{opt}}}(G)\le \frac{5α(G)-1}{2}$. Moreoevr, for the case $α(G)\leq 3$, we improve the upper bound to $4$, which is tight.

math.CO↗

arXiv2020

Stable structure on safe set problems in vertex-weighted graphs

Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=\sum_{v \in S} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $\mathrm{s}(G,w)$ and connected weighted safe number $\mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. Note that for every pair $(G,w)$, $\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$ by their definitions. Recently, it was asked which pair $(G,w)$ satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs $G$ such that $\mathrm{s}(G,w)=\mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.

math.CO↗

arXiv2019

Safe sets in digraphs

A non-empty subset $S$ of the vertices of a digraph $D$ is called a {\it safe set} if \begin{itemize} \item[(i)] for every strongly connected component $M$ of $D-S$, there exists a strongly connected component $N$ of $D[S]$ such that there exists an arc from $M$ to $N$; and \item[(ii)] for every strongly connected component $M$ of $D-S$ and every strongly connected component $N$ of $D[S]$, we have $|M|\leq |N|$ whenever there exists an arc from $M$ to $N$. \end{itemize} In the case of acyclic digraphs a set $X$ of vertices is a safe set precisely when $X$ is an {\it in-dominating set}, that is, every vertex not in $X$ has at least one arc to $X$. We prove that, even for acyclic digraphs which are traceable (have a hamiltonian path) it is NP-hard to find a minimum cardinality in-dominating set. Then we show that the problem is also NP-hard for tournaments and give, for every positive constant $c$, a polynomial algorithm for finding a minimum cardinality safe set in a tournament on $n$ vertices in which no strong component has size more than $c\log{}(n)$. Under the so called Exponential Time Hypothesis (ETH) this is close to best possible in the following sense: If ETH holds, then, for every $ε>0$ there is no polynomial time algorithm for finding a minimum cardinality safe set for the class of tournaments in which the largest strong component has size at most $\log^{1+ε}(n)$. We also discuss bounds on the cardinality of safe sets in tournaments.

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arXiv2019

On sufficient conditions for rainbow cycles in edge-colored graphs

Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges of $G$ and the number of colors appearing on $E(G)$, respectively. For a vertex $v\in V(G)$, the \emph{color neighborhood} of $v$ is defined as the set of colors assigned to the edges incident to $v$. A subgraph of $G$ is \emph{rainbow} if all of its edges are assigned with distinct colors. The well-known Mantel's theorem states that a graph $G$ on $n$ vertices contains a triangle if $e(G)\geq\lfloor\frac{n^2}{4}\rfloor+1$. Rademacher (1941) showed that $G$ contains at least $\lfloor\frac{n}{2}\rfloor$ triangles under the same condition. Li, Ning, Xu and Zhang (2014) proved a rainbow version of Mantel's theorem: An edge-colored graph $G$ has a rainbow triangle if $e(G)+c(G)\geq n(n+1)/2$. In this paper, we first characterize all graphs $G$ satisfying $e(G)+c(G)\geq n(n+1)/2-1$ but containing no rainbow triangles. Motivated by Rademacher's theorem, we then characterize all graphs $G$ which satisfy $e(G)+c(G)\geq n(n+1)/2$ but contain only one rainbow triangle. We further obtain two results on color neighborhood conditions for the existence of rainbow short cycles. Our results improve a previous theorem due to Broersma, Li, Woeginger, and Zhang (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles.

math.CO↗

arXiv2019

Optimal proper connection of graphs

An edge-colored graph $G$ is called properly colored if no two adjacent edges share a color in $G$. An edge-colored connected graph $G$ is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. In this paper, we discuss how to make a connected graph properly connected efficiently. More precisely, we consider the problem to convert a given monochromatic graph into properly connected by recoloring $p$ edges with $q$ colors so that $p+q$ is as small as possible. We discuss how this can be done efficiently for some restricted graphs, such as trees, complete bipartite graphs and graphs with independence number $2$.

math.CO↗

arXiv2018

On the weighted safe set problem on paths and cycles

Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $s(G,w)$ and connected weighted safe number $cs(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. It is easy to see that for any pair $(G,w)$, ${s}(G,w) \le {cs}(G,w)$ by their definitions. In this paper, we discuss the possible equality when $G$ is a path or a cycle. We also give an answer to a problem due to Tittmann et al. [Eur. J. Combin. Vol. 32 (2011)] concerning subgraph component polynomials for cycles and complete graphs.

math.CO↗

arXiv2018

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common vertex. It is not known whether every 3 longest paths in a connected graph have a common vertex and similarly for 4, 5, and 6 longest path. In [5] the authors give an upper bound on distance among 3 longest paths in a connected graph. In this paper we give a similar upper bound on distance between 4 longest paths and also for k longest paths, in general.

math.CO↗

arXiv2017

Kernels by properly colored paths in arc-colored digraphs

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.

math.CO↗

arXiv2017

Decomposing edge-colored graphs under color degree constraints

For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then $V(G)$ can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least $2$. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.

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