Shinya Fujita’s research while affiliated with Yokohama City University and other places

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}, $\gamma_h(G)$, 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 $\delta(G)\ge 2$, then $\gamma_h(G)\le \frac{2n}{5}$, and the bound is tight. We also give some tight upper bounds on $\gamma_h(G)$ for {triangle-free} graphs G that contain a Hamiltonian path or a Hamiltonian cycle.

Let t, k, d be integers with 1≤d≤k-1$1 \le d \le k - 1$ and t≥8d+11$t \ge 8d + 11$, and let n=k(t+1)$n = k(t+1)$. We show that if G is a graph of order n such that δ(G)≥d$\delta (G) \ge d$ and |E(G)|≥n2-(d+1)n+dt+12(d2+3d+4)$|E(G)| \ge \left( {\begin{array}{c}n\\ 2\end{array}}\right) - (d + 1)n + dt + \frac{1}{2}(d^{2} + 3d + 4)$, then G has a K1,t$K_{1, t}$-factor. We also give a construction showing the sharpness of the condition on |E(G)|.

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) = ℓ(G), where ℓ(G) is the number of vertices of a longest path in G. Another function we introduce is χPOC(G; t) giving the minimum number of colors required over all weightings of G using t distinct weights. We show that the ratio of χPOC(G; t) − 1 to χ(G) − 1 can be bounded by t for any graph G; in fact, the result is shown by determining χPOC(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.

An edge-colored connected graph G is properly connected if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced the optimal proper connection number pcopt(G) for a monochromatic connected graph G, to make a connected graph properly connected efficiently. More precisely, pcopt (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 pcopt (G) has an upper bound in terms of the independence number α(G). Namely, we prove that for a connected graph G, pcopt (G)≤5α(G)−12. Moreover, for the case α(G)≤3, we improve the upper bound to 4, which is tight.

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 $\chi_{POC}(G;t)$ giving the minimum number of colors required over all weightings of G using t distinct weights. We show that the ratio of $\chi_{POC}(G;t)-1$ to $\chi(G)-1$ can be bounded by t for any graph G; in fact, the result is shown by determining $\chi_{POC}(G;t)$ 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.

Let G be a graph, and let w be a non-negative real-valued weight function on V(G). For every subset X of V(G), let w(X)=∑v∈Xw(v). A non-empty subset S⊂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)≥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 numbers(G,w) and connected weighted safe numbercs(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 every pair (G, w), s(G,w)≤cs(G,w) by their definitions. In [Journal of Combinatorial Optimization, 37:685–701, 2019], the authors asked which pair (G, w) satisfies the equality s(G,w)=cs(G,w) and it was shown that every weighted cycle satisfies the equality. In the companion paper [European Journal of Combinatorics, in press] of this paper, we give a complete list of connected bipartite graphs G such that s(G,w)=cs(G,w) for every weight function w on V(G). In this paper, as is announced in the companion paper, we show that, for any graph G in this list and for any weight function w on V(G), there exists an FPTAS for calculating a minimum connected safe set of (G, w). In order to prove this result, we also prove that for any tree T and for any weight function w′ on V(T), there exists an FPTAS for calculating a minimum connected safe set of (T,w′). This gives a complete answer to a question posed by Bapat et al. [Networks, 71:82–92, 2018] and disproves a conjecture by Ehard and Rautenbach [Discrete Applied Mathematics, 281:216–223, 2020]. We also show that determining whether a graph is in the above list or not can be done in linear time.

An edge-colored graph G is properly colored if no two adjacent edges share a color in G. An edge-colored connected graph G is 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.

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)=∑v∈Sw(v). A non-empty subset S⊂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)≥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. Note that for every pair (G,w), s(G,w)≤cs(G,w) by their definitions. In [Fujita et al. “On weighted safe set problem on paths and cycles” J. Comb. Optim. 37 (2019) 685–701], 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 s(G,w)=cs(G,w) for every weight function w on V(G).