Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. For any given integer $k\ge 1$, a graph $G$ is $(P_2\cup kP_1)$ free if $G$ does not contain the disjoint union of $P_2$ and $k$ isolated vertices as an induced subgraph.
In this note, we show that every 4-tough and $2k$-connected $(P_2\cup kP_1)$-free graph with at least three vertices is hamiltonian.
This result in some sense is an "extension" of the classical Chvátal-Erdős Theorem that every $\max\{2,k\}$-connected
$(k+1)P_1$-free graph on at least three vertices is hamiltonian.
The toughness of a noncomplete graph $G$ is the maximum real number $t$ such that the ratio of $|S|$ to the number of components of $G-S$ is at least $t$ for every cutset $S$ of $G$, and the toughness of a complete graph is defined to be $\infty$. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the d…
Let $k \geq 2$ be an integer. We say that a graph $G$ is $(K_2 \cup kK_1)$-free if it does not contain $K_2 \cup kK_1$ as an induced subgraph. Recently, Shi and Shan conjectured that every $1$-tough and $2k$-connected $(K_2 \cup kK_1)$-free graph is hamiltonian. In this paper, we solve this conjecture by proving the statement; every $1$-tough and $k$-connected $(K_2 \cup kK_1)$-free graph with minimum degree at least $\frac{3(k-1)}{2}$ is hamiltonian or the Petersen graph.
Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, resp…
Chvátal conjectured in 1973 the existence of some constant $t$ such that all $t$-tough graphs with at least three vertices are hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in general. We say that a graph is $(K_2 \cup 3K_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup 3K_1$, where $K_2 \cup 3K_1$ is the disjoint union of an edge and three isolated vertices. In this paper, we show that every 3-tough $(K_2 \cup 3K_1)$-free graph with at least three vertices is hamiltonian.
Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. For a positive real number $t$, a non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. A complete graph is said to be $t$-tough for any $t>0$. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is Hamiltonian, is still open in general. Chvátal and Erdös \cite{CE} proved that, for any integer $k\ge 1$, every $\max\…
A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ is a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free graph has a 2-factor. The toughness condition of this result is sharp. Moreover, for any $\varepsilon>0$ there exists a $(2-\varepsilon)$-tough $2P_5$-free graph without a 2-factor. This implies that the graph $P_4 \cup P_{10}$ is best possible for a forbidden subgraph in a sense.
In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ e…
The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal's conjecture from another perspective: what is the spectral condition to guarantee the existence of a hamiltonian cycle among $t$-tough graphs? We first give the answer to $1$-tough graphs, i.e. if $ρ(G)\geqρ(M_{n})$, then $G$ contains a hamiltonian cycle, unless $G\cong M_{n}$, where $M_{n}=K_{1}\nabla K_{n-4}^{+3}$ and $K_{n-4}^{+3}$ is the graph obtained from $3K_{1}\cup K_{n-4}$ by adding three independent edges betwee…
For a 1-tough graph $G$ we define $σ_3(G) = \min\{°(u) + °(v)+ °(w):$ $\{u, v, w\}$ is an independent set of vertices$\}$ and $NC2(G)=\min \{|N(u)\cup N(v)|: d(u,v)=2\}$. D. Bauer, G. Fan and H.J.Veldman proved that $c(G)\geq \min\{n,2NC2(G)\}$ for any 1-tough graph $G$ with $σ_3(G)\geq n\geq 3$, where $c(G)$ is the circumference of $G$ (D. Bauer, G. Fan and H.J.Veldman,Hamiltonian properties of graphs with large neighborhood unions,Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: $c(G)\geq\min\{n,2NC2(G)+4\}$.In this paper, we prove this conject…
A pair of orthogonal one-factorizations $\mathcal{F}$ and $\mathcal{G}$ of the complete graph $K_n$ is totally $C_4$-free, if the union $F\cup G$, for any $F,G\in\mathcal{F}\cup\mathcal{G}$, does not include a cycle of length four. In this note, we prove if $q\equiv3$ (mod 4) is a prime power with $q\geq11$, then there is a pair of orthogonal totally $C_4$-free one-factorizations of $K_{q+1}$.
Let $Oct_{1}^{+}$ and $Oct_{2}^{+}$ be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For $Oct_{1}^{+}$, we prove that a 4-connected graph is $Oct_{1}^{+}$-free if and only if it is $C_{6}^{2}$, $C_{2k+1}^{2}$ $(k \geq 2)$ or it is obtained from $C_{5}^{2}$ by repeatedly 4-splitting vertices. We also show that a planar graph is $Oct_{1}^{+}$-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from $\{K_{1}, K_{2} ,K_{3}\} \cup \mathscr{K} \cup \{Oct,L_{5} \}$, where $\mathscr{K}$ is the set of graphs obtai…
Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this note we prove that $G$ contains a heterochromatic cycle of length 4 if $G$ has $n\geq 60$ vertices and $|CN(u)\cup CN(v)|\geq n-1$ for every pair of vertices $u$ and $v$ of $G$. This extends a result of Broersma et al. on the existence of heterochromatic cycles of length 3 or 4.
Given an infinite group $G$ and a subset $A$ of $G$ we let $Δ(A) = {g \in G : |gA \cap A| =\infty}$ (this is sometimes called the combinatorial derivation of $A$). A subset $A$ of $G$ is called large if there exists a finite subset $F$ of $G$ such that $FA=G$. We show that given a large set $X$, and a decomposition $X=A_1 \cup ... \cup A_n$, there must exist an $i$ such that $Δ(A_i)$ is large. This answers a question of Protasov. We also answer a number of related questions of Protasov.
Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\ldots,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\cup B_i$ and $B_i\cup A_{i+1}$ (subscripts module $t$) are edges for all $i=0,1,\ldots,t-1$. Let $\mathcal{H}$ be a $k$-uniform $n$-vertex hypergraph with $n\geq 5k$ and $n$ divisible by $k$. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists $α\in (0,1)$ …
Let $G$ be a simple $n$-vertex graph with degree sequence $d_1,d_2,...,d_n$ and vertex set $\V(G)$. The degree of $v\in\V(G)$ is denoted by $\D(v)$. The smallest integer $r$ for which $\V(G)$ has an $r$-partition $$ \V(G)=V_1\cup V_2\cup...\cup V_r,\quad V_i\cap V_j=\emptyset, \quad,i\neq j $$ such that $\D(v)\leq n-\abs{V_i}$, $\forall v\in V_i$, $i=1,2,...,r$ is denoted by $\f(G)$. In this note we prove the inequality $$ \f(G)\geq\frac n{n-\bar{\bar{d}}}, $$ where $\bar{\bar{d}}=\sqrt{\dfrac{d_1^2+d_2^2+...+d_n^2}n}$.
The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and $E(G)$ is the edge set of $G$. In this note, a sharp lower bound of $A\!^*$ in terms of the maximum degree for the case of trees is derived. The $n$-vertex trees having maximal and minimal $A\!^*$ values are also characterized here. Moreover, it is shown that $A\!^*\!(G)$ is non-negative even integer for every graph $G$ and that there exist infinitely many connected gr…
A non-isolated vertex $x\in V(G)$ is called $C_{3}$-free if $x$ belongs to no triangle of $G$. In \cite{BMW} Burger, Mynhardt and Weakley introduced the idea of universal fixers. Let $G=(V,E)$ be a graph with $n$ vertices and $G'$ a copy of $G$. For a bijective function $π:V(G)\mapsto V (G')$, we define the prism $πG$ of $G$ as follows: $V(πG)=V(G)\cup V(G')$ and $E(πG)=E(G)\cup E(G')\cup M_π$, where $M_π=\{uπ(u): u\in V(G)\}$. Let $γ(G)$ be the domination number of $G$. If $γ(πG)=γ(G)$ for any bijective function $π$, then $G$ is called a universal fixer. In \cite{MX} it is conjectured that th…
The Grundy domination number of a simple graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$, $v_i \in V$, that satisfies the property $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Here, $N(v) = \{u : uv \in E\}$ and $N[v] = N(v) \cup \{v\}$. In this note, we prove a recent conjecture about the Grundy domination number of the strong product of two graphs. We then discuss how this result relates to the zero forcing number of the strong product of graphs.
The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to th…
A note on cycles in cyclically $4$-edge-connected cubic planar graphs
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Tongji University（同济大学数学科学学院，智能计算与应用教育部重点实验室，同济大学）
The $k$-rainbow independent domination number of a graph $G$, denoted $γ_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in $V(G)\setminus (V_1\cup V_2)$ has neighbors in both $V_1$ and $V_2$. This domination invariant was proposed by {Š}umenjak, Rall and Tepeh in (Applied Mathematics and Computation 333(15), 2018: 353-361), which allows to reduce the problem of computing the independent domination number of the generalized prism $G {\Box} K_k$ to an integer labeling problem on $G$. They proved a Nord…
A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and no vertex of $B$. Few explicit families of $n$-e.c. are known for $n > 2$. In this short note, we give a new construction of 3-e.c. graphs using the notion of quadrance in the finite Euclidean space $\mathbbm{Z}_p^d$.
In this note we consider the bent linear 2-tree and provide an explicit formula for the resistance distance $r_{G_n}(1,n)$ between the first and last vertices of the graph. We call the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$ a straight linear 2-tree. We define the graph $H_n$ with $V(H_n)= V(G_n)$ and $E(H_n) = (E(G_n) \cup \{k,k+3\})\setminus(\{k+1,k+3\}$ to be a bent linear 2-tree with bend at vertex $k$.
The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from $X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and show that the resulting graph witnesses that $b(2,t) =Ω\left(t^{3/2}/\log t \right)$, thereby improving the best known …
Let $G$ be a directed graph such that the in-degree of any vertex $G$ is at least one. Let also ${\mathcalτ}: V(G)\rightarrow \Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset $M$ of vertices of $G$ is called a dynamic monopoly for $(G,τ)$ if the vertex set of $G$ can be partitioned into $D_0\cup... \cup D_t$ such that $D_0=M$ and for any $i\geq 1$ and any $v\in D_i$, the number of edges from $D_0\cup... \cup D_{i-1}$ to $v$ is at least $τ(v)$. One of the most applicable and widely studied threshold assignments in directed graphs is strict majority threshold assignment i…
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger's conjecture and the Erdős-Lovász Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $k\ge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $ω(G)<χ(G)=s+t-1$ satisfie…
The game of cops and robber is a turn based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say the team of cops win the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in each component of a graph is called the cop number of the graph. Sivaraman [Discrete Math. 342(2019), pp. 2306-2307] conjectured that for every $t\geq 5$, the cop number of a connected $P_t$-free graph is at most $t-3$, where $P_t$ denotes a path on…
