Vertex Colouring Edge Weightings: a Logarithmic Upper Bound on Weight-Choosability

A graph $G$ is said to be $(k,m)$-choosable if for any assignment of $k$-element lists $L_v \subset \mathbb{R}$ to the vertices $v \in V(G)$ and any assignment of $m$-element lists $L_e \subset \mathbb{R}$ to the edges $e \in E(G)$ there exists a total weighting $w: V(G) \cup E(G) \rightarrow \mathbb{R}$ of $G$ such that $w(v) \in L_v$ for any vertex $v \in V(G)$ and $w(e) \in L_e$ for any edge $e \in E(G)$ and furthermore, such that for any pair of adjacent vertices $u,v$, we have $w(u)+ \sum_{e \in E(u)}w(e) \neq w(v)+ \sum_{e \in E(v)}w(e)$, where $E(u)$ and $E(v)$ denote the edges incident to $u$ and $v$ respectively. In this paper we give an algorithmic proof showing that any graph $G$ without isolated edges is $(1, 2 \lceil \log_2(\Delta(G)) \rceil+1)$-choosable, where $\Delta(G)$ denotes the maximum degree in $G$.

Sistem Pendukung Keputusan Program Kerja Pengawasan Tahunan Menggunakan Metode Simple Additive Weighting

The Annual Supervision Work Program (ASWP) is a measure of efficiency and effectiveness in implementing supervision in supporting Inspectorate performance. For the preparation of this ASWP using a Risk-Based Supervision Planning (RBSP) based on certain criteria. Decision Support System (DSS) is a system used to determine alternative ASWP to be implemented. The criteria are determined using the Simple Additive Weighting (SAW) method. The basic concept of the SAW method is to find the total weighting of the performance rating for each alternative. Testing is done from ASWP alternative normalized according to the type of attribute criteria (benefit or cost). The final result is obtained from the calculation process, namely from the 10 test data there is 80% compatibility of the data from the system calculation. The sum of the normalized matrix with weights per criterion shows an alternative ranking of the regional apparatus organization that is closest to the criteria to the most distant from the criteria. In order to get an alternative ASWP to be implemented.

Sistem Pendukung Keputusan Program Kerja Pengawasan Tahunan Menggunakan Metode Simple Additive Weighting (Studi Kasus di Inspektorat Kabupaten Rokan Hulu)

The Annual Supervision Work Program (ASWP) is a measure of efficiency and effectiveness in implementing supervision in supporting Inspectorate performance. For the preparation of this ASWP using a Risk-Based Supervision Planning (RBSP) based on certain criteria. Decision Support System (DSS) is a system used to determine alternative ASWP to be implemented. The criteria are determined using the Simple Additive Weighting (SAW) method. The basic concept of the SAW method is to find the total weighting of the performance rating for each alternative. Testing is done from ASWP alternative normalized according to the type of attribute criteria (benefit or cost). The final result is obtained from the calculation process, namely from the 10 test data there is 80% compatibility of the data from the system calculation. The sum of the normalized matrix with weights per criterion shows an alternative ranking of the regional apparatus organization that is closest to the criteria to the most distant from the criteria. In order to get an alternative ASWP to be implemented.

Permanent Index of Matrices Associated with Graphs

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $\phi_f(u) \ne \phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. This approach leads to conjectures on the permanent indices of matrices $A_G$ and $B_G$ associated to a graph $G$. In this paper, we establish a method that reduces the study of permanent of matrices associated to a graph $G$ to the study of permanent of matrices associated to induced subgraphs of $G$. Using this reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable.

A Note on Neighbour-Distinguishing Regular Graphs Total-Weighting

We investigate the following modification of a problem posed by Karoński, Łuczak and Thomason [J. Combin. Theory, Ser. B 91 (2004) 151–157]. Let us assign positive integers to the edges and vertices of a simple graph $G$. As a result we obtain a vertex-colouring of $G$ by sums of weights assigned to the vertex and its adjacent edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary $G$? We know that the answer is yes if $G$ is a 3-colourable, complete or 4-regular graph. Moreover, it is enough to use weights from $1$ to $11$, as well as from $1$ to $\lfloor{\chi(G)\over2}\rfloor+1$, for an arbitrary graph $G$. Here we show that weights from $1$ to $7$ are enough for all regular graphs.