ترغب بنشر مسار تعليمي؟ اضغط هنا

On the number of proper paths between vertices in edge-colored hypercubes

76   0   0.0 ( 0 )
 نشر من قبل Yang Weihua
 تاريخ النشر 2017
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Given an integer $1leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1leq i leq j$, have color $1$ and all other edges have color $2$. Cheng et al. [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] determined the number of distinct shortest properly colored paths between a pair of vertices for the $(1)$-colored hypercubes. It is natural to consider the number for $(j)$-coloring, $jgeq 2$. In this note, we determine the number of different shortest proper paths in $(j)$-colored hypercubes for arbitrary $j$.



قيم البحث

اقرأ أيضاً

115 - Minjia Shi 2020
A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4$-by-...-by-$4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $F^4(4;2,2)$, find a testing set of size $25$ for $F^3(4;2,2)$, and derive an upper bound on the number of $F^n(4;2,2)$. Additionally, for any $n$ greater than $2$, we construct an $F^n(4;2,2)$ that cannot be refined to a latin hypercube, while each of its sub-$F^{n-1}(4;2,2)$ can. Keywords: frequency hypercube, frequency square, latin hypercube, testing set, MDS code
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $delta^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $vin V(G)$. A cycle of $(G,c)$ is called proper if any t wo adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $ngeq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $ell$ for every $ell$ with $3 le ell le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $ngeq 3$ vertices with $delta^c(G)geq frac{n+1}{2}$ is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that $(G,c)$ does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.
Let $mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(mathcal{C}, t)$-saturated if $G$ does not contain any graph in $mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $mat hcal{C}$. Similarly to classical saturation functions, define $mathrm{sat}_t(n, mathcal{C})$ to be the minimum number of edges in a $(mathcal{C},t)$ saturated graph. Let $mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors. In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for $mathrm{sat}_t(n, mathcal{C}_r(K_k))$ for all $r$, showing a sharp change in behavior when $rgeq binom{k-1}{2}+2$. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine $mathrm{sat}_t(n, mathcal{C}_2(K_3))$ exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.
Let $F$ be a fixed graph. The rainbow Turan number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$ all of whose e dges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraete. In this paper, we show that the rainbow Turan number of a path with $k+1$ edges is less than $left(frac{9k}{7}+2right) n$, improving an earlier estimate of Johnston, Palmer and Sarkar.
It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Delta^{mon}(G)leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker result s for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا