Browsing by Subject "cactus graphs"

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Access status: Open Access ,
Bounds on the 2-domination number in cactus graphs
(2006) Chellali, Mustapha
A $2$-dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex not in $S$ is dominated at least twice. The minimum cardinality of a $2$-dominating set of $G$ is the $2$-domination number $\gamma_{2}(G)$. We show that if $G$ is a nontrivial connected cactus graph with $k(G)$ even cycles ($k(G)\geq 0$), then $\gamma_{2}(G)\geq\gamma_{t}(G)-k(G)$, and if $G$ is a graph of order n with at most one cycle, then $\gamma_{2}(G)\geqslant(n+\ell-s)/2$ improving Fink and Jacobson's lower bound for trees with $\ell>s$, where $\gamma_{t}(G)$, $\ell$ and $s$ are the total domination number, the number of leaves and support vertices of $G$, respectively. We also show that if $T$ is a tree of order $n\geqslant 3$, then $\gamma_{2}(T)\leqslant\beta(T)+s-1$, where $\beta(T)$ is the independence number of $T$.
Access status: Open Access ,
Classical solutions of initial problems for quasilinear partial functional differential equations of the first order
(2006) Czernous, Wojciech
We consider the initial problem for a quasilinear partial functional differential equation of the first order $\partial_t z(t,x)+\sum_{i=1}^nf_i(t,x,z_{(t,x)})\partial_{x_i} z(t,x)=G(t,x,z_{(t,x)}),\\ z(t,x)=\varphi(t,x)\;\;((t,x)\in[-h_0,0]\times R^n)$ where $z_{(t,x)}\colon\,[-h_0,0]\times[-h,h]\to R$ is a function defined by $z_{(t,x)}(\tau,\xi)=z(t+\tau,x+\xi)$ for $(\tau,\xi)\in[-h_0,0]\times[-h,h]$. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.
Access status: Open Access ,
Local irregularity conjecture for 2-multigraphs versus cacti
(Wydawnictwa AGH, 2024) Grzelec, Igor; Woźniak, Mariusz
A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. A locally irregular colorable multigraph $G$ is any multigraph which admits a locally irregular coloring. We denote by $\textrm{lir}(G)$ the locally irregular chromatic index of a multigraph $G$, which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph $G$. In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph $G$, which is not isomorphic to $K_2$, multigraph $^{2}G$ obtained from $G$ by doubling each edge satisfies $\textrm{lir}(^2G)\leq 2$. We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph $G$ satisfies $\textrm{lir}(G)\leq 3$. At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved.