A totally silver coloring of a graph G is a k--coloring of G such that for every vertex v \in V(G), each color appears exactly once on N[v], the closed neighborhood of v. A totally silver graph is a graph which admits a totally silver coloring. Totally silver coloring are directly related to other areas of graph theory such as distance coloring and domination. In this work, we present several constructive characterizations of totally silver graphs and bipartite totally silver graphs. We give several infinite families of totally silver graphs. We also give cubic totally silver graphs of girth up to 10.
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 $K_{q+1}$.
A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free W…
We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.
Let the edges of the complete graph $K_n$ be coloured red or blue, and let $G$ be a graph with $|V(G)| < n$. Then ot(n,G) is defined to be the minimum integer, if it exists, such that any such colouring of $K_n$ contains a copy of $G$ with $r$ red edges and $b$ blue edges for any $r,b \geq 0$ with $r+b= e(G)$. If ot(n,G) exists for every sufficiently large $n$, we say that $G$ is \emph{omnitonal}. Omnitonal graphs were introduced by Caro, Hansberg and Montejano [arXiv:1810.12375,2019]. Now let $G_1$, $G_2$ be two copies of $G$ with their edges coloured red or blue. If there is a colour-preserv…
A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($Δ(G)+2$)-coloring, where $Δ(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $Δ(G) \geq 9$ or $Δ(G) \in \{7, 8\}$ with some restrictions has a total $(Δ(G) + 1)$-colori…
Total coloring is a variant of edge coloring where both vertices and edges are to be colored. A graph is totally $k$-choosable if for any list assignment of $k$ colors to each vertex and each edge, we can extract a proper total coloring. In this setting, a graph of maximum degree $Δ$ needs at least $Δ+1$ colors. In the planar case, Borodin proved in 1989 that $Δ+2$ colors suffice when $Δ$ is at least 9. We show that this bound also holds when $Δ$ is $8$.
Clemson University（克莱姆森大学）；University of Alaska, Fairbanks（阿拉斯加大学费尔班克斯分校）；Framingham State University（弗兰明翰州立大学）；George Mason University（乔治梅森大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；The Ohio State University（俄亥俄州立大学）；Columbia University（哥伦比亚大学）
Amalgamation in the totally non-negative part of positroid varieties is equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and identify total non-negativity via edge signatures. Here we provide an explicit characterization of such signatures on the planar bicolored trivalent directed perfect networks in the disk parametrizing positroid cells $S_M^{TNN}$.To a graph $G$ representing $S_M^{TNN}$, we associate a geometric signature satisfying full rank condition …
Triangulations and soliton graphs for totally positive Grassmannian
KP方程是一个非线性色散波动方程，它为浅水波的共振相互作用提供了一个很好的模型。KP方程的正则孤子解可以由完全非负的Grassmannian Gr $(N,M)_{\geq 0}$中的点构造。Kodama和Williams研究了KP孤子的渐近模式（热带极限），称为孤子图，并表明它们对应于Postnikov的Le图。在本文中，我们考虑孤立子图的KP族，家庭的交换流是兼容的KP方程。对于正的Grassmannian Gr $(2,M)_{>0}$，Kodama和Williams证明了孤子图与$M$ -边形的三角剖分是双射的.当$N=3$和$M=6,7$和$8$同时存在时，我们将这个结果推广到Gr $(N,M)_{>0}$.在每一种情况下，我们表明，孤子图是在双射与Postnikov的plabic图，它推广了乐图。
Let $H$ be a directed acyclic graph other than a rooted star. It is known that there are constants $c(H)$ and $C(H)$ such that the following holds for the complete directed graph $D_n$. There are at most $C\log n$ directed acyclic subgraphs of $D_n$ that cover every $H$-copy of $D_n$, while fewer than $c\log n$ directed acyclic subgraphs of $D_n$ do not cover all $H$-copies. Here this dichotomy is considerably strengthened. Let ${\vec G}(n,p)$ denote the random directed graph. The {\em fractional arboricity} of $H$ is $a(H) = max \{\frac{|E(H')|}{|V(H')|-1}\}$, where the maximum is over all no…
Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total colorings together for designing new types of graph homomorphisms: totally-colored graph homomorphisms, graphic-lattice homomorphisms from sets to sets, every-zero graphic group homomorphisms from sets to sets. Our graph-homomorphism lattices are made up by graph homomorphisms. These new homomorphisms induce some problems of graph theory, for example, Number Strin…
Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well-known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in one-twelfth of a hexagon with free boundary to express them as perfect matchings of a family of non-bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit …
The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicolored (plabic) graphs in the disk. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations on vectors on such graphs, depending on some signatures defined on edges. The problem of characterizing the signatures corresponding to the totally non-negative cells, was left open in [38]. In our paper we provide an explicit construction of such signatures, satisfying both the…
Optimal realisations of two-dimensional, totally-decomposable metrics
度量$d$在有限集$X$上的实现是加权图$(G,w)$，其顶点集包含$X$，使得被认为是$G$中的顶点的$X$的元素之间的最短路径距离等于$d$。这样的实现$(G,w)$被称为最优的，如果它的边缘权重的总和是最小的在所有这样的实现。最优实现总是存在的，尽管一般来说计算它们是NP-难的，并且它们在遗传学，电气网络和互联网断层扫描等领域有应用。在[Adv. in Math.53，1984，321-402] A.~ Dress表明，度量$d$的最优实现与某种多面体复形密切相关，该多面体复形可以正则地与$d$相关联，称为其紧跨度。此外，他强调，（加权）图组成的零和一维面的紧跨度的$d$必须始终包含一个最佳实现作为同胚子图。在本文中，我们证明了这一猜想确实适用于某一类度量，即一类完全”=可分解的度量，其紧跨度有二维。作为推论，最小曼哈顿网络问题是寻找二维全可分解度量的最优实现的特殊情况。
Let $G=(V,E)$ be a graph with no isolated vertices. A vertex $v$ totally dominate a vertex $w$ ($w \ne v$), if $v$ is adjacent to $w$. A set $D \subseteq V$ called a total dominating set of $G$ if every vertex $v\in V$ is totally dominated by some vertex in $D$. The minimum cardinality of a total dominating set is the total domination number of $G$ and is denoted by $γ_t(G)$. A total dominator coloring of graph $G$ is a proper coloring of vertices of $G$, so that each vertex totally dominates some color class. The total dominator chromatic number $χ_{td}(G)$ of $G$ is the least number of color…
