arXiv subjects

Morteza Hasanvand

Publications and source records attributed to Morteza Hasanvand.

At least 19 recordsLinked to original sources

arXiv2025

On the existence of minimally tough graphs having large minimum degrees

Kriesel conjectured that every minimally $1$-tough graph has a vertex with degree precisely $2$. Katona and Varga (2018) proposed a generalized version of this conjecture which says that every minimally $t$-tough graph has a vertex with degree precisely $\lceil 2t\rceil$, where $t$ is a positive real number. This conjecture has been recently verified for several families of graphs. For example, Ma, Hu, and Yang (2023) confirmed it for claw-free minimally $3/2$-tough graphs. Recently, Zheng and Sun (2024) disproved this conjecture by constructing a family of $4$-regular graphs with toughness approaching to $1$. In this paper, we disprove this conjecture for planar graphs and their line graphs. In particular, we construct an infinite family of minimally $t$-tough non-regular claw-free graphs with minimum degree close to thrice their toughness. This construction not only disproves a renewed version of Generalized Kriesel's Conjecture on non-regular graphs proposed by Zheng and Sun (2024), it also gives a supplement to a result due to Ma, Hu, and Yang (2023) who proved that every minimally $t$-tough claw-free graph with $t\ge 2$ has a vertex of degree at most $3t+ \lceil (t-5)/3\rceil$. Moreover, we conjecture that there is not a fixed constant $c$ such that every minimally $t$-tough graph has minimum degree at most $\lceil c t \rceil$.

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arXiv2025

The List Square Coloring Conjecture fails for bipartite planar graphs and their line graphs

Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several authors whether this conjecture holds for bipartite graphs with small degrees, claw-free graphs, or line graphs. In this paper, we introduce several kinds of counterexamples to this conjecture to solve three open problems posed by Kim and Park~(2015), Kim, Kwon, and Park~(2015), and Dai, Wang, Yang, and Yu~(2018). In particular, we disprove a planar version of this conjecture proposed by Havet, Heuvel, McDiarmid, and Reed (2017). This conjecture was originally proposed to make a stronger version of the List Total Coloring Conjecture. In order to make a revised version, it remains to decide whether this conjecture holds for bipartite graphs $G$ by imposing a lower bound on the chromatic number of the square graph $G^2$ in terms of its maximum degree as the condition $χ(G^2) \ge \frac{1}{2} Δ(G^2)+1$ (or by adding an upper bound on the number of colors used in lists for a weaker version). To support this version, we will show that the bipartite condition cannot be dropped even by increasing the lower bound arbitrarily. Finally, we investigate non-choosable graphs with bounded maximum degree in bipartite or planar graphs. Consequently, we improve several graph constructions due to Erd\H os, Rubin, and Taylor~(1980), Bessy, Havet, and Palaysi (2002), Voigt (1993), Mirzakhani (1996), and Glebov, Kostochka, and Tashkinov (2005) in terms of maximum degree or order. In addition, we characterize edge-minimal $3$-chromatic non-$3$-choosable (resp. $4$-chromatic non-$4$-choosable) graphs of order at most $9$ (resp. $11$) and settle a question posed by Nelsen~(2019).

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arXiv2024

Edge-decompositions of $O(m)$-edge-connected graphs into isomorphic copies of a fixed tree of size $m$

In this paper, we show that every $O(m)$-edge-connected simple graph $G$ of size divisible by $m$ with minimum degree at least $2^{O(m)}$ has an edge-decomposition into isomorphic copies of any given tree $T$ of size $m$. Moreover, the minimum degree condition can be dropped for graphs $G$ with girth greater than the diameter of $T$. These results improve two results due to Bensmail, Harutyunyan, Le, Merker, and Thomassé (2017) and Merker (2017) who gave a factorial upper bound on the necessary edge-connectivity.

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arXiv2024

Equitable factorizations of highly edge-connected graphs: complete characterizations

In this paper, we show that every highly edge-connected graph $G$, under a necessary and sufficient degree condition, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$ with $1\le i\le k$, $|d_{G_i}(v)-d_G(v)/k|<1$. This characterization covers graphs having at least $k-1$ vertices with degree not divisible by $k$. In addition, we investigate almost equitable factorizations in arbitrary edge-connected graphs. Next, we establish a simpler criterion for the existence of factorizations $G_1,\ldots, G_k$ satisfying $d_{G_i}(v)\ge \lfloor d_G(v)/k\rfloor$ for all vertices $v$ (reps. $d_{G_i}(v)\le \lceil d_G(v)/k\rceil$). As an application, we come up with a criterion to determine whether a highly edge-connected graph with $δ(G)\ge δ_1+\cdots+ δ_m$ (resp. $Δ(G)\le Δ_1+\cdots+ Δ_m$) can be edge-decomposed into factors $G_1,\ldots, G_m$ satisfying $δ(G_i)\ge δ_i$ (resp. $Δ(G_i)\le Δ_i$) for all $i$ with $1\le i \le m$, provided that $δ_1+\cdots+ δ_m$ is divisible by an odd number $p$ and $δ_i\ge p-1\ge 2$ (resp. $Δ_1+\cdots+ Δ_m$ is divisible by $p$ and $Δ_i\ge p-1\ge 2$). For graphs of even order, we replace an odd-edge-connectivity condition. In particular, for the special case $m=2$, we refine the needed odd-edge-connectivity further by giving a sufficient odd-edge-connectivity condition for a graph $G$ to have a partial parity factor $F$ such that for each vertex $v$ with a given parity constraint, $| d_{F}(v)-\varepsilon d_G(v)|< 2$, and for all other vertices $v$, $| d_{F}(v)-\varepsilon d_G(v)|\le 1$, where $\varepsilon $ is a real number and $0< \varepsilon < 1$. Finally we introduce another application on the existence of almost even factorizations of odd-edge-connected graphs.

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arXiv2024

Spanning tree-connected subgraphs with small degrees

Let $G$ be a graph with a spanning subgraph $F$, let $m$ be a positive integer, and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S\subseteq V(G)$, $$Ω_m(G\setminus S)\le \sum_{v\in S}\big(f(v)-2m\big)+m+Ω_m(G[S]),$$ then $G$ has a spanning $m$-tree-connected subgraph $H$ containing $F$ such that for each vertex $ v$, $d_H(v)\le f(v)+\max\{0,d_F(v)-m\}$, where $G[S]$ denotes the induced subgraph of $G$ with the vertex set $S$ and $Ω_m(G_0)$ is a parameter to measure $m$-tree-connectivity of a given graph $G_0$. By applying this result, we show that every $k$-edge-connected graph $G$ with $k\ge 2m$ has a spanning $m$-tree-connected subgraph $H$ such that $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$ for each $v\in V(H)$; moreover, if $G$ is $k$-tree-connected and $k\ge m$, then $G$ has a spanning $m$-tree-connected subgraph $H$ such that $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m$ for each $v\in V(H)$. As a consequence, we conclude that every $(r-2m)$-edge-connected graph with $r\ge 4m$ admits a spanning $m$-tree-connected subgraph with maximum degree at most $3m$. Next, we prove that a graph $G$ admits a spanning $m$-tree-connected subgraph $H$ satisfying $Δ(H) \le 2m+1$, if for all $S\subseteq V(G)$, $$ ω(G\setminus S)+\small {\frac{m+1}{2}}\, iso(G\setminus S) \le \frac{1}{m}|S|+1,$$ where $ω(G\setminus S)$ and $iso(G\setminus S)$ denote the number of components and the number of isolated vertices of $G\setminus S$, respectively. As a consequence, we conclude that every $m(n-1)$-connected $K_{1, n}$-free simple graph with a sufficiently large minimum degree and $n\ge 3$ admits a spanning $m$-tree-connected subgraph with maximum degree at most $2m+1$.

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arXiv2024

Modulo factors with bounded degrees

Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive integer, and let $f:V(G)\rightarrow \{-1,\ldots, k-2\}$ be a mapping with $\sum_{v\in X}f(v) \stackrel{k}{\equiv}\sum_{v\in Y}f(v)$. In this paper, we show that if $G$ is essentially $(3k-3)$-edge-connected and for each vertex $v$, $d_G(v)\ge 2k-1+f(v)$, then it admits a factor $H$ such that for each vertex $v$, $d_H(v)\stackrel{k}{\equiv} f(v)$, and $$\lfloor\frac{d_G(v)}{2}\rfloor-(k-1)\le d_{H}(v)\le \lceil\frac{d_G(v)}{2}\rceil+k-1.$$ Next, we generalize this result to general graphs and derive sufficient conditions for a highly edge-connected general graph $G$ to have a factor $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),f(v)+k\}$. Finally, we show that every $(4k-1)$-edge-connected essentially $(6k-7)$-edge-connected graph admits a bipartite factor whose degrees are positive and divisible by $k$.

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arXiv2022

The existence of tree-connected $\{g,f\}$-factors in edge-connected graphs and tough graphs

In 1970 Lov{á}sz gave a necessary and sufficient condition for the existence of a factor $F$ in a graph $G$ such that for each vertex $v$, $g(v)\le d_F(v)\le f(v)$, where $g$ and $f$ are two integer-valued functions on $V(G)$ with $g\le f$. In this paper, we give a sufficient edge-connectivity condition for the existence of an $m$-tree-connected factor $H$ in a bipartite graph $G$ with bipartition $(X,Y)$ such that its complement is $m_0$-tree-connected and for each vertex $v$, $d_H(v)\in \{g(v),f(v)\}$, provided that for each vertex $v$, $g(v)+m_0\le \frac{1}{2}d_G(v)\le f(v)-m$ and $|f(v)-g(v)|\le k$, and there is $h(v)\in \{g(v),f(v)\}$ in which $\sum_{v\in X}h(v)=\sum_{v\in Y}h(v)$. Moreover, we generalize this result to general graphs. As an application, we give sufficient conditions for the existence of tree-connected $\{g,f\}$-factors in edge-connected graphs and tough graphs.

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arXiv2022

Toughness and the existence of tree-connected $\{f,f+k\}$-factors

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$ satisfying $2m\le f\le b$, where $b$ and $m$ are two positive integers with $b\ge 4m^2$. In this paper, we show that if $G$ is $b^2$-tough and $|V(G)|\ge b^2$, then it has an $m$-tree-connected factor $H$ such that for each vertex $v$, $$d_H(v)\in \{f(v), f(v)+1\}.$$ Next, we generalize this result by giving sufficient conditions for a tough graph to have a tree-connected factors $H$ such that for each vertex $v$, $d_H(v)\in \{f(v), f(v)+k\}$. As an application, we prove that every $64b(b-a)^2$-tough graph $G$ of order at least $b+1$ with $ab|V(G)|$ even admits a connected factor whose degrees lie in the set $\{a,b\}$, where $a$ and $b$ are two integers with $2\le a< b < \frac{6}{5}a$. Moreover, we prove that every $16$-tough graph $G$ of order at least three admits a $2$-connected factor whose degrees lie in the set $\{2,3\}$, provided that $G$ has a $2$-factor with girth at least five. This result confirms a weaker version of a long-standing conjecture due to Chvátal (1973).

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arXiv2022

A necessary and sufficient condition for the existence of $\{p,p+1,q-1,q\}$-orientations in simple graphs

Let $G$ be a simple graph and let $p$ and $q$ be two integer-valued functions on $V(G)$ with $p< q$ in which for each $v\in V(G)$, $q(v) \ge \frac{1}{2}d_G(v)$ and $p(v) \ge \frac{1}{2} q(v)-2$. In this note, we show that $G$ has an orientation such that for each vertex $v$, $d^+_G(v)\in\{p(v),p(v)+1,q(v)-1,q(v)\}$ if and only if it has an orientation such that for each vertex $v$, $p(v) \le d^+_G(v)\le q(v)$ where $d^+_G(v)$ denotes the out-degree of $v$ in $G$. From this result, we refine a result due to Addario-Berry, Dalal, and Reed (2008) in bipartite simple graphs on the existence of degree constrained factors.

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arXiv2022

Highly tree-connected complementary modulo factors with bounded degrees

Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive integer, and let $f:V(G)\rightarrow Z_k$ be a mapping with $\sum_{v\in X}f(v) \stackrel{k}{\equiv}\sum_{v\in Y}f(v)$. In this paper, we show that if $G$ is $(2m+2m_0+4k-4)$-edge-connected and $m+m_0>0$, then $G$ has an $m$-tree-connected factor $H$ such that its complement is $m_0$-tree-connected and for each vertex $v$, $d_H(v)\stackrel{k}{\equiv} f(v)$, and $$\lfloor\frac{d_G(v)}{2}\rfloor-(k-1)-m_0\le d_{H}(v)\le \lceil\frac{d_G(v)}{2}\rceil+k-1+m.$$ Next, we generalize this result to general graphs and derive a sufficient degree condition for a highly edge-connected general graph $G$ to have a connected factor $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),f(v)+k\}$. Finally, we show that every $(4k-2)$-tree-connected graph admits a bipartite connected factor whose degrees are divisible by $k$.

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arXiv2022

The existence of $\{p,q\}$-orientations in edge-connected graphs

In 1976 Frank and Gy{á}rf{á}s gave a necessary and sufficient condition for the existence of an orientation in an arbitrary graph $G$ such that for each vertex $v$, the out-degree $d^+_G(v)$ of it satisfies $p(v)\le d^+_G(v)\le q(v)$, where $p$ and $q$ are two integer-valued functions on $V(G)$ with $p\le q$. In this paper, we give a sufficient edge-connectivity condition for the existence of an orientation in $G$ such that for each vertex $v$, $d^+_G(v)\in \{p(v),q(v)\}$, provided that for each vertex $v$, $p(v)\le \frac{1}{2}d_G(v) \le q(v)$, $|q(v)-p(v)|\le k$, and there is $t(v)\in \{p(v),q(v)\}$ in which $|E(G)|=\sum_{v\in V(G)}t(v)$. This result is a generalization of a theorem due to Thomassen (2012) on the existence of modulo orientations in highly edge-connected graphs.

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arXiv2022

The existence of planar $4$-connected essentially $6$-edge-connected graphs with no claw-decompositions

In 2006 Bar{á}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with edge-connectivity $4$ which the smallest one contains $24$ vertices. In this note, we first give a smaller counterexample having only $18$ vertices and next construct a family of planar $4$-connected essentially $6$-edge-connected $4$-regular simple graphs of size divisible by three with no claw-decompositions. This result provides the sharpness for two known results which say that every $5$-edge-connected graph of size divisible by three admits a claw-decomposition if it is essentially $6$-edge-connected or planar.

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arXiv2022

Modulo orientations with bounded out-degrees

Let $G$ be a graph, let $k$ be a positive integer, and let $p:V(G)\rightarrow Z_k$ be a mapping with $|E(G)| \stackrel{k}{\equiv}\sum_{v\in V(G)}p(v) $. In this paper, we show that if $G$ is $(3k-3)$-edge-connected, then it has an orientation such that for each vertex $v$, $|d^+_G(v)-d_G(v)/2| < k$; also if $G$ contains $2k-2$ edge-disjoint spanning trees, then it admits such an orientation but by imposing greater out-degree bounds.

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arXiv2022

Tutte's $3$-Flow Conjecture in $3$-tree-connected graphs

Tutte's $3$-flow conjecture says that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. Kochol (2001) showed that it is enough to prove this conjecture for $5$-edge-connected graphs. Former, Jaeger, Linial, Payan, and Tarsi (1992) conjectured that every $5$-edge-connected graph is $Z_3$-connected and so it admits a nowhere-zero $3$-flow. In this note, we show that if the second conjecture would be true, then every $3$-tree-connected graph must also be $Z_3$-connected and so Tutte's $3$-flow conjecture can be extended to this family of graphs.

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arXiv2022

The existence of $m$-tree-connected $(g,f+f'-m)$-factors using $(g,f)$-factors and $m$-tree-connected $(m,f')$-factors

Let $G$ be a graph and let $g$, $f$, and $f'$ be three positive integer-valued functions on $V(G)$ with $g\le f$. Tokuda, Xu, and Wang (2003) showed that if $G$ contains a $(g,f)$-factor and a spanning $f'$-tree, then $G$ also contains a connected $(g,f+f'-1)$-factor. In this note, we develop their result to a tree-connected version by proving that if $G$ contains a $(g,f)$-factor and an $m$-tree-connected $(m,f')$-factor, then $G$ also contains an $m$-tree-connected $(g,f+f'-m)$-factor, provided that $f\ge m$. In addition, we show that $g$ allows to be nonnegative.

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arXiv2022

Spanning trees and spanning closed walks with small degrees

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S\subseteq V(G)$, $ω(G\setminus S)<\sum_{v\in S}(f(v)-2)+2+ω(G[S])$, then $G$ has a spanning tree $T$ containing an arbitrary given matching such that for each vertex $v$, $d_T(v)\le f(v)$, where $ω(G\setminus S)$ denotes the number of components of $G\setminus S$ and $ω(G[S])$ denotes the number of components of the induced subgraph $G[S]$ with the vertex set $S$. This is an improvement of several results. Next, we prove that if for all $S\subseteq V(G)$, $ω(G\setminus S)\le \sum_{v\in S} (f(v)-1)+1$, then $G$ admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex $v$ at most $f(v)$ times. This result solves a long-standing conjecture due to Jackson and Wormald (1990).

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arXiv2022

Factors and connected factors in tough graphs with high isolated toughness

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. Assume that for all $S\subseteq V(G)$, $$\sum_{v\in I(G\setminus S)}f(v)(f(v)+1)\le |S|,$$ where $I(G\setminus S)$ denotes the set of isolated vertices of $G\setminus S$. In this paper, we show that if for all $S\subseteq V(G)$, $$ω(G\setminus S)\le \sum_{v\in S}(f(v)-1)+1,$$ and $\sum_{v\in V(G)}f(v)$ is even, then $G$ has a factor $F$ such that for each vertex $v$, $d_F(v)=f(v)$, where $ω(G\setminus S)$ denotes the number of components of $G\setminus S$. Moreover, we show that if for all $S\subseteq V(G)$, $$ω(G\setminus S)\le \frac{1}{4}|S|+1,$$ and $f\ge 2$, then $G$ has a connected factor $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),f(v)+1\}$.

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arXiv2021

Equitable factorizations of edge-connected graphs

In this paper, we show that every $(3k-3)$-edge-connected graph $G$, under a certain condition on whose degrees, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_G(v)/k|< 1$, where $1\le i\le k$. As application, we deduce that every $6$-edge-connected graph $G$ can be edge-decomposed into three factors $G_1$, $G_2$, and $G_3$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_{G}(v)/3|< 1$, unless $G$ has exactly one vertex $z$ with $d_G(z) \stackrel{3}{\not\equiv}0$. Next, we show that every odd-$(3k-2)$-edge-connected graph $G$ can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $d_{G_i}(v)$ and $d_G(v)$ have the same parity and $|d_{G_i}(v)-d_G(v)/k|< 2$, where $k$ is an odd positive integer and $1\le i\le k$. Finally, we give a sufficient edge-connectivity condition for a graph $G$ to have a parity factor $F$ with specified odd-degree vertices such that for each vertex $v$, $| d_{F}(v)-\varepsilon d_G(v)|< 2$, where $\varepsilon $ is a real number with $0< \varepsilon < 1$.

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