arXiv subjects

Kenta Ozeki

Publications and source records attributed to Kenta Ozeki.

33 records · Page 2Linked to original sources

arXiv2022

On Reachable Assignments under Dichotomous Preferences

We consider the problem of determining whether a target item assignment can be reached from an initial item assignment by a sequence of pairwise exchanges of items between agents. In particular, we consider the situation where each agent has a dichotomous preference over the items, that is, each agent evaluates each item as acceptable or unacceptable. Furthermore, we assume that communication between agents is limited, and the relationship is represented by an undirected graph. Then, a pair of agents can exchange their items only if they are connected by an edge and the involved items are acceptable. We prove that this problem is PSPACE-complete even when the communication graph is complete (that is, every pair of agents can exchange their items), and this problem can be solved in polynomial time if an input graph is a tree.

cs.GT↗

arXiv2022

Reforming an Envy-Free Matching

We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

cs.GT↗

arXiv2021

Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams

We initiate the study of $k$-edge-connected orientations of undirected graphs through edge flips for $k \geq 2$. We prove that in every orientation of an undirected $2k$-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is $k$-edge-connected. This yields an ``edge-flip based'' new proof of Nash-Williams' theorem: an undirected graph $G$ has a $k$-edge-connected orientation if and only if $G$ is $2k$-edge-connected. As another consequence of the theorem, we prove that the edge-flip graph of $k$-edge-connected orientations of an undirected graph $G$ is connected if $G$ is $(2k+2)$-edge-connected. This has been known to be true only when $k=1$.

math.CO↗

arXiv2020

Single-conflict colouring

Given a multigraph, suppose that each vertex is given a local assignment of $k$ colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least $k$ for which this is always possible given any set of local assignments we call the {\em single-conflict chromatic number} of the graph. This parameter is closely related to separation choosability and adaptable choosability. We show that single-conflict chromatic number of simple graphs embeddable on a surface of Euler genus $g$ is $O(g^{1/4}\log g)$ as $g\to\infty$. This is sharp up to the logarithmic factor.

math.CO↗

arXiv2019

The Alon-Tarsi number of $K_5$-minor-free graphs

In this paper, we show the following three theorems. Let $G$ be a $K_5$-minor-free graph. Then Alon-Tarsi number of $G$ is at most $5$, there exists a matching $M$ of $G$ such that the Alon-Tarsi number of $G-M$ is at most $4$, and there exists a forest $F$ such that the Alon-Tarsi number of $G-E(F)$ is at most $3$.

math.CO↗

arXiv2018

On the minimum leaf number of cubic graphs

The \emph{minimum leaf number} $\hbox{ml} (G)$ of a connected graph $G$ is defined as the minimum number of leaves of the spanning trees of $G$. We present new results concerning the minimum leaf number of cubic graphs: we show that if $G$ is a connected cubic graph of order $n$, then $\mathrm{ml}(G) \leq \frac{n}6 + \frac13$, improving on the best known result in [Inf. Process. Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph Theory and Applications 5 (2017) 207-211]. We further prove that if $G$ is also 2-connected, then $\mathrm{ml}(G) \leq \frac{n}{6.53}$, improving on the best known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new conjectures concerning the minimum leaf number of several types of cubic graphs and examples showing that the bounds of the conjectures are best possible.

math.CO↗

arXiv2017

Decomposing planar cubic graphs

The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.

math.CO↗

arXiv2017

A note on a Brooks' type theorem for DP-coloring

Dvořák and Postle \cite{DP} introduced a \textit{DP-coloring} of a simple graph as a generalization of a list-coloring. They proved a Brooks' type theorem for a DP-coloring, and Bernshteyn, Kostochka and Pron \cite{BKP} extended it to a DP-coloring of multigraphs. However, detailed structure when a multigraph does not admit a DP-coloring was not specified in \cite{BKP}. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.

math.CO↗

arXiv2017

A Sufficient condition for DP-4-colorability

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each $k \in \{3, 4, 5, 6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results.

math.CO↗

arXiv2017

On Homeomorphically Irreducible Spanning Trees in Cubic Graphs

A spanning tree without a vertex of degree two is called a Hist which is an abbreviation for homeomorphically irreducible spanning tree. We provide a necessary condition for the existence of a Hist in a cubic graph. As one consequence, we answer affirmatively an open question on Hists by Albertson, Berman, Hutchinson and Thomassen.

math.CO↗

arXiv2017

Sufficient conditions for the existence of a path-factor which are related to odd components

In this paper, we are concerned with sufficient conditions for the existence of a $\{P_{2},P_{2k+1}\}$-factor. We prove that for $k\geq 3$, there exists $\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\sum_{0\leq j\leq k-1}c_{2j+1}(G-X)\leq \varepsilon_{k}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{2k+1}\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$. On the other hand, we construct infinitely many graphs $G$ having no $\{P_{2},P_{2k+1}\}$-factor such that $\sum_{0\leq j\leq k-1}c_{2j+1}(G-X)\leq \frac{32k+141}{72k-78}|X|$ for all $X\subseteq V(G)$.

math.CO↗

arXiv2016

Hamiltonicity of planar graphs with a forbidden minor

Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not extend to $K_{2,5}$-minor-free $3$-connected graphs in general, as shown by the Petersen graph, and does not extend to $K_{2,6}$-minor-free $3$-connected planar graphs, as we show by an infinite family of examples.

math.CO↗

arXiv2016

On two conjectures about the proper connection number of graphs

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of $G$ are connected by at least one proper path in $G$. In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if $G$ is a noncomplete graph with connectivity $κ(G) = 2$ and minimum degree $δ(G)\ge 3$, then $pc(G) = 2$, posed by Borozan et al.~in [Discrete Math. 312(2012), 2550-2560]. We give a family of counterexamples to disprove this conjecture. However, from a result of Thomassen it follows that 3-edge-connected noncomplete graphs have proper connection number 2. Using this result, we can prove that if $G$ is a 2-connected noncomplete graph with $diam(G)=3$, then $pc(G) = 2$, which solves the second conjecture we want to mention, posed by Li and Magnant in [Theory \& Appl. Graphs 0(1)(2015), Art.2].

math.CO↗

arXiv2016

A characterization of $K_{2,4}$-minor-free graphs

We provide a complete structural characterization of $K_{2,4}$-minor-free graphs. The $3$-connected $K_{2,4}$-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains $K_4$ and, for each $n \ge 5$, $2n-8$ nonisomorphic graphs of order $n$. To describe the $2$-connected $K_{2,4}$-minor-free graphs we use $xy$-outerplanar graphs, graphs embeddable in the plane with a Hamilton $xy$-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, $xy$-outerplanar graphs are precisely the graphs that have no rooted $K_{2,2}$-minor where $x$ and $y$ correspond to the two vertices on one side of the bipartition of $K_{2,2}$. Each $2$-connected $K_{2,4}$-minor-free graph is then (i) outerplanar, (ii) the union of three $xy$-outerplanar graphs and possibly the edge $xy$, or (iii) obtained from a $3$-connected $K_{2,4}$-minor-free graph by replacing each edge $x_iy_i$ in a set $\{x_1 y_1, x_2 y_2, \ldots, x_k y_k\}$ satisfying a certain condition by an $x_i y_i$-outerplanar graph.

math.CO↗

arXiv2015

A New Approach Towards a Conjecture on Intersecting Three Longest Paths

In 1966, T. Gallai asked whether every connected graph has a vertex that appears in all longest paths. Since then this question has attracted much attention and many work has been done in this topic. One important open question in this area is to ask whether any three longest paths contains a common vertex in a connected graph. It was conjectured that the answer to this question is positive. In this paper, we propose a new approach in view of distances among longest paths in a connected graph, and give a substantial progress towards the conjecture along the idea.

math.CO↗