arXiv subjects

Toru Hasunuma

Publications and source records attributed to Toru Hasunuma.

6 recordsLinked to original sources

arXiv2025

Mader's Conjecture and Its Variants for Cographs

The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of $P_4$-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree $T$ of order $m$, every $k$-connected cograph $G$ with $\delta(G) \geq \left\lfloor \frac{3k}{2} \right\rfloor +m-1$ contains a subtree $T' \cong T$ such that $G-V(T')$ is still $k$-connected, where $\delta(G)$ denotes the minimum degree of $G$. Moreover, we show that three variants of Mader's conjecture hold for cographs, that is, for any tree $T$ of order $m$, $\bullet$ every $k$-connected (respectively, $k$-edge-connected) cograph $G$ with $\delta(G) \geq k+m-1$ contains a subtree $T' \cong T$ such that $G-E(T')$ is $k$-connected (respectively, $k$-edge-connected), $\bullet$ every $k$-edge-connected cograph $G$ with $\delta(G) \geq k+m-[k = 1]$ contains a subtree $T' \cong T$ such that $G-V(T')$ is $k$-edge-connected, where we use Iverson's convention for $[k = 1]$. We furthermore present tight lower bounds on the minimum degree of a cograph for the existence of disjoint connectivity keeping trees, a maximal connectedness keeping tree and a super edge-connectedness keeping tree.

math.CO↗

arXiv2025

Exponentiation of Graphs

Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that every connected exponential graph is maximally connected. For exponential graphs, we also present a necessary and sufficient condition to be super edge-connected and sufficient conditions to be Hamiltonian and to have edge-disjoint Hamiltonian cycles and completely independent spanning trees. Applying our results to previously known networks, we have maximally connected and super edge-connected Hamiltonian graphs of doubly exponential order with logarithmic diameter. We furthermore define iterated exponential graphs which may be of not only practical but also theoretical interest.

math.CO↗

arXiv2023

Connectivity Preserving Hamiltonian Cycles in $k$-Connected Dirac Graphs

We show that for $k \geq 2$, there exists a function $f(k) = O(k)$ such that every $k$-connected graph $G$ of order $n \geq f(k)$ with minimum degree at least $\frac{n}{2}$ contains a Hamiltonian cycle $H$ such that $G-E(H)$ is $k$-connected. Applying Nash-Williams' result on edge-disjoint Hamiltonian cycles, we also show that for $k \geq 2$ and $\ell \geq 2$, there exists a function $g(k,\ell) = O(k\ell)$ such that every $k$-connected graph $G$ of order $n \geq g(k,\ell)$ with minimum degree at least $\frac{n}{2}$ contains $\ell$ edge-disjoint Hamiltonian cycles $H_1,H_2,\ldots,H_\ell$ such that $G-\cup_{1 \leq i \leq \ell}E(H_i)$ is $k$-connected. As a corollary, we have a statement that refines the result of Nash-Williams for $k$-connected graphs with $k \leq 8$. Moreover, when the connectivity of $G$ is exactly $k$, a similar result with an improved lower bound on $n$ can be shown, which does not depend on the result of Nash-Williams.

math.CO↗

arXiv2022

Completely Independent Spanning Trees in Line Graphs

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees in $L(K_n)$ where the number $\lfloor \frac{n+1}{2} \rfloor$ is optimal, such that $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees still exist in the graph obtained from $L(K_n)$ by deleting any vertex (respectively, any induced path of order at most $\frac{n}{2}$) for $n = 4$ or odd $n \geq 5$ (respectively, even $n \geq 6$). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where $δ(G)$ denotes the minimum degree of $G$. $\ $ $\bullet$ Every $2k$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is not super edge-connected or $δ(G) \geq 2k$. $\ $ $\bullet$ Every $(4k-2)$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is regular. $\ $ $\bullet$ Every $(k^2+2k-1)$-connected line graph $L(G)$ with $δ(G) \geq k+1$ has $k$ completely independent spanning trees.

math.CO↗

arXiv2010

A linear time algorithm for L(2,1)-labeling of trees

An L(2,1)-labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ and $y$ are adjacent and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are at distance 2, for all $x$ and $y$ in $V(G)$. A $k$-L(2,1)-labeling is an assignment $f:V(G)\to\{0,..., k\}$, and the L(2,1)-labeling problem asks the minimum $k$, which we denote by $λ(G)$, among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of a very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been $\mO(Δ^{4.5} n)$ for more than a decade, however, an $\mO(n^{1.75})$-time algorithm has been proposed recently, which substantially improved the previous one, where $Δ$ is the maximum degree of $T$ and $n=|V(T)|$. In this paper, we finally establish a linear time algorithm for L(2,1)-labeling of trees.

cs.DS↗

arXiv2009

A tight upper bound on the (2,1)-total labeling number of outerplanar graphs

A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident to $x$, and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are a pair of adjacent vertices or a pair of adjacent edges, for all $x$ and $y$ in $V(G)\cup E(G)$. The $(2,1)$-total labeling number $λ^T_2(G)$ of a graph $G$ is defined as the minimum $k$ among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155, 2585--2593 (2007)], Chen and Wang conjectured that all outerplanar graphs $G$ satisfy $λ^T_2(G) \leq Δ(G)+2$, where $Δ(G)$ is the maximum degree of $G$, while they also showed that it is true for $G$ with $Δ(G)\geq 5$. In this paper, we solve their conjecture completely, by proving that $λ^T_2(G) \leq Δ(G)+2$ even in the case of $Δ(G)\leq 4 $.

cs.DM↗