Search ResultsShowing 1-17 of 17
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The existence of a $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$-factor based on the size or the $A_α$-spectral radius of graphs
Comments: 20 pagesCategories: math.COLet $G$ be a connected graph of order $n$. A $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$-factor of $G$ is a spanning subgraph of $G$ such that each component is isomorphic to a member in $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$, where $\mathcal{T}(3)$ is a $\{1,2,3\}$-tree. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. In this paper, we obtain a lower bound on the size or the $A_{\alpha}$-spectral radius for $\alpha\in[0,1)$ of $G$ to guarantee that $G$ has a $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$-factor, and construct an extremal graph to show that the bound on $A_{\alpha}$-spectral radius is optimal.
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A spectral condition for a graph having a strong parity factor
Comments: 11 pagesJournal: Discrete Applied Mathematics 360(2025)188-195Categories: math.COKeywords: strong parity factor, spectral condition, spectral radius condition, connected graph contains, spanning subgraphTags: journal articleA graph $G$ contains a strong parity factor $F$ if for every subset $X\subseteq V(G)$ with $|X|$ even, $G$ has a spanning subgraph $F$ satisfying $\delta(F)\geq1$, $d_F(u)\equiv1$ (mod 2) for any $u\in X$, and $d_F(v)\equiv0$ (mod 2) for any $v\in V(G)\setminus X$. In this paper, we give a spectral radius condition to guarantee that a connected graph contains a strong parity factor.
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arXiv:2308.07653 (Published 2023-08-15)
Connectivity Graph-Codes
Categories: math.COThe symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$ of spanning subgraphs of $H$ a connectivity code for $H$ if the symmetric difference of any two distinct subgraphs in ${\cal G}$ is a connected spanning subgraph of $H$. It is easy to see that the maximum possible cardinality of such a collection is at most $2^{k'(H)} \leq 2^{\delta(H)}$, where $k'(H)$ is the edge-connectivity of $H$ and $\delta(H)$ is its minimum degree. We show that equality holds for any $d$-regular (mild) expander, and observe that equality does not hold in several natural examples including powers of long cycles and products of a small clique with a long cycle.
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arXiv:2305.16108 (Published 2023-05-25)
A spectral radius condition for a graph to have $(a,b)$-parity factors
Let $a,b$ be two positive integers such that $a \le b$ and $a \equiv b$ (mod $2$). We say that a graph $G$ has an $(a,b)$-parity factor if $G$ has a spanning subgraph $F$ such that $d_{F}(v) \equiv b$ (mod $2$) and $a \le d_{F}(v) \le b$ for all $v \in V (G)$. In this paper, we provide a tight spectral radius condition for a graph to have $(a,b)$-parity factors.
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arXiv:2204.04373 (Published 2022-04-09)
Tight toughness, isolated toughness and binding number bounds for the $\{K_2,C_n\}$-factors
Categories: math.COThe $\{K_2,C_n\}$-factor of a graph is a spanning subgraph whose each component is either $K_2$ or $C_n$. In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the $\{K_2,C_{2i+1}| i\geq 2 \}$-factor for any graph is obtained, which answers a problem due to Gao and Wang (J. Oper. Res. Soc. China (2021), https://doi.org/10.1007/s40305-021-00357-6).
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arXiv:2108.02685 (Published 2021-08-05)
Irregular Subgraphs
Categories: math.COWe suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $1$. The second is that every graph on $n$ vertices with minimum degree $\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{\delta+1}+1$. Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$. In particular we show that if $d^3 \log n \leq o(n)$ then every $d$-regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$. We also prove that any graph with $n$ vertices and minimum degree $\delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{\delta+1}+2$ times.
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arXiv:2103.11304 (Published 2021-03-21)
The fullerenes with a perfect star packing
Comments: 19pages,6figuresCategories: math.COA spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if every component of the spanning subgraph is isomorphic to the star graph $K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of $G$ such that each vertex of $G$ not in $D$ is adjacent to exactly one vertex from $D$ and any two vertices of $D$ are not adjacent in $G$. Fullerene graph is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. Clearly, a perfect star packing in a fullerene graph $G$ on $n$ vertices will exist if and only if $G$ has an efficient dominating set of cardinality $\frac{n}{4}$. The problem of finding an efficient dominating set is algorithmically hard \cite{Alg_hard}. In this paper, we give a characterization for a fullerene graph to own a perfect star packing. And mainly show that it is necessary for a fullerene $G$ owning a perfect star packing to have order being divisible by $8$. This answers an open problem asked by Dosli\'{c} et. al. and also shows that a fullerene graph with an efficient dominating set has $8n$ vertices. By the way, we find some counterexamples for the necessity of Theorem $14$ in \cite{Doslic} and list some forbidden configurations to preclude the existence of a perfect star packing of type $P0$.
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arXiv:2004.01289 (Published 2020-04-02)
Weak saturation numbers of complete bipartite graphs in the clique
Comments: 15 pages, 3 figuresCategories: math.COThe notion of weak saturation was introduced by Bollob\'as in 1968. Let $F$ and $H$ be graphs. A spanning subgraph $G \subseteq F$ is weakly $(F,H)$-saturated if it contains no copy of $H$ but there exists an ordering $e_1,\ldots,e_t$ of $E(F)\setminus E(G)$ such that for each $i \in [t]$, the graph $G \cup \{e_1,\ldots,e_i\}$ contains a copy $H'$ of $H$ such that $e_i \in H'$. Define $wsat(F,H)$ to be the minimum number of edges in a weakly $(F,H)$-saturated graph. In this paper, we prove for all $t \ge 2$ and $n \ge 3t-3$, that $wsat(K_n,K_{t,t}) = (t-1)(n + 1 - t/2)$, and we determine the value of $wsat(K_n,K_{t-1,t})$ as well. For fixed $2 \le s < t$, we also obtain bounds on $wsat(K_n,K_{s,t})$ that are asymptotically tight.
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arXiv:2001.09168 (Published 2020-01-24)
The Threshold Dimension of a Graph
Comments: 27 pages, 14 FiguresCategories: math.COLet $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$, is the cardinality of a smallest set $W$ of vertices such that every pair of vertices of $G$ is resolved by some vertex of $W$. The threshold dimension of a graph $G$, denoted $\tau(G)$, is the minimum metric dimension among all graphs $H$ having $G$ as a spanning subgraph. In other words, the threshold dimension of $G$ is the minimum metric dimension among all graphs obtained from $G$ by adding edges. If $\beta(G) = \tau(G)$, then $G$ is said to be \emph{irreducible}; otherwise, we say that $G$ is reducible. If $H$ is a graph having $G$ as a spanning subgraph and such that $\beta(H)=\tau(G)$, then $H$ is called a threshold graph of $G$. The threshold dimension of a graph is expressed in terms of a minimum number of strong products of paths that admits a certain type of embedding of the graph. A sharp upper bound for the threshold dimension of trees is established. It is also shown that the irreducible trees are precisely those of metric dimension at most 2. Moreover, if $T$ is a tree with metric dimension 3 or 4, then $T$ has threshold dimension $2$. It is shown, in these two cases, that a threshold graph for $T$ can be obtained by adding exactly one or two edges to $T$, respectively. However, these results do not extend to trees with metric dimension $5$, i.e., there are trees of metric dimension $5$ with threshold dimension exceeding $2$.
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arXiv:1806.00135 (Published 2018-05-31)
Packing spanning partition-connected subgraphs with small degrees
Categories: math.COLet $G$ be a graph with $X\subseteq V(G)$ and let $l$ be an intersecting supermodular subadditive integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. Let $\lambda \in [0,1]$ be a real number and let $\eta $ be a real function on $X$. In this paper, we show that if $G$ is $l$-partition-connected and for all $S\subseteq X$, $$\Theta_l(G \setminus S) \le \sum_{v\in S} (\eta(v) -2l(v))+l(V(G))+l(S)-\lambda (e_G(S))+l(S)),$$ then $G$ has an $l$-partition-connected spanning subgraph $H$ such that for each vertex $v\in X$, $d_H(v)\le \lceil \eta(v) -\lambda l(v) \rceil $, where $e_G(S)$ denotes the number of edges of $G$ with both ends in $S$ and $\Theta_l(G \setminus S)$ denotes the maximum number of all $\sum_{A\in P} l(A)-e_{G\setminus S}(P)$ taken over all partitions $P$ of $V(G)\setminus S$. Finally, we show that if $H$ is an $(l_1+\cdots +l_m)$-partition-connected graph, then it can be decomposed into $m$ edge-disjoint spanning subgraphs $H_1,\ldots, H_m$ such that every graph $H_i$ is $l_i$-partition-connected, where $l_1, l_2,\ldots, l_m$ are $m$ intersecting supermodular subadditive integer-valued functions on subsets of $V(H)$. These results generalize several known results.
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arXiv:1606.03790 (Published 2016-06-13)
The super spanning connectivity of arrangement graph
Categories: math.COA $k$-container $C(u, v)$ of a graph $G$ is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u, v)$ of $G$ is a $k^*$-container if it is a spanning subgraph of $G$. A graph $G$ is $k^*$-connected if there exists a $k^*$-container between any two different vertices of G. A $k$-regular graph $G$ is super spanning connected if $G$ is $i^*$-container for all $1\le i\le k$. In this paper, we prove that the arrangement graph $A_{n, k}$ is super spanning connected if $n\ge 4$ and $n-k\ge 2$.
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arXiv:1408.5488 (Published 2014-08-23)
Saturation in the Hypercube and Bootstrap Percolation
Comments: 21 pages, 2 figuresLet $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. Given graphs $F$ and $H$, a spanning subgraph $G$ of $F$ is said to be weakly $(F,H)$-saturated if the edges of $E(F)\setminus E(G)$ can be added to $G$ one at a time so that each additional edge creates a new copy of $H$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.
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arXiv:1301.7356 (Published 2013-01-30)
Fractional Perfect b-Matching Polytopes. I: General Theory
Comments: 37 pagesThe fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number b_v. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.
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Domination game played on trees and spanning subgraphs
Comments: 16 pages, 9 figuresJournal: Discrete Mathematics 313(2013) 915-923Categories: math.COKeywords: spanning subgraph, staller-start game domination number, domination game, optimal strategy-dominator plays, staller playsTags: journal articleThe domination game is played on a graph G. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of G dominated to that point in the game. Both players use an optimal strategy---Dominator plays so as to end the game as quickly as possible while Staller plays in such a way that the game lasts as many steps as possible. The game domination number of G is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number of G when Staller starts the game. In this paper these two games are studied when played on trees and spanning subgraphs. A lower bound for the game domination number of a tree in terms of the order and maximum degree is proved and shown to be asymptotically tight. It is shown that for every k, there is a tree T with game domination number k and Staller-start game domination number k+1, and it is conjectured that there is no tree with game domination number k and Staller-start game domination number k-1. A relation between the game domination number of a graph and its spanning subgraphs is considered. It is proved that for any positive integer n, there exists a graph G and its spanning tree T such that the game domination number of G is at least n more than the game domination number of T. Moreover, there exist 3-connected graphs G having a spanning subgraph such that the game domination number of the spanning subgraph is arbitrarily smaller than that of G.
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arXiv:0707.0227 (Published 2007-07-02)
Uniformly Weighted Star-Factors of Graphs
Categories: math.COA {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. An {\it edge-weighting} of $G$ is a function $w: E(G)\longrightarrow \mathbb{N}^+$, where $\mathbb{N}^+$ is the set of positive integers. Let $\Omega$ be the family of all graphs $G$ such that every star-factor of $G$ has the same weights under a fixed edge-weighting $w$. In this paper, we present a simple structural characterization of the graphs in $\Omega$ that have girth at least five.
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arXiv:0707.0224 (Published 2007-07-02)
Uniform Star-factors of Graphs with Girth Three
Categories: math.COA {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. Recently, Hartnell and Rall studied a family $\mathscr{U}$ of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family $\mathscr{U}$ when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine all members of this family.
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The chromatic numbers of double coverings of a graph
Comments: 10 pagesJournal: Discrete Math. 308(2008) 5078-5086Categories: math.COSubjects: 05C15Tags: journal articleIf we fix a spanning subgraph $H$ of a graph $G$, we can define a chromatic number of $H$ with respect to $G$ and we show that it coincides with the chromatic number of a double covering of $G$ with co-support $H$. We also find a few estimations for the chromatic numbers of $H$ with respect to $G$.