‎‎let $g=(v,e)$ be a simple graph‎. ‎an edge labeling $f:eto {0,1}$ induces a vertex labeling $f^+:vtoz_2$ defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$‎, ‎where $z_2={0,1}$ is the additive group of order 2‎. ‎for $iin{0,1}$‎, ‎let‎ ‎$e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$‎. ‎a labeling $f$ is called edge-friendly if‎ ‎$|e_f(1)-e_f(0)|le 1$‎. ‎$i_f(g)=v_f(...

an h-magic labeling in a h-decomposable graph g is a bijection f : v (g) ∪ e(g) → {1, 2, ..., p + q} such that for every copy h in the decomposition, σνεv(h) f(v) +  σeεe(h) f(e) is constant. f is said to be h-e-super magic if f(e(g)) = {1, 2, · · · , q}. a family of subgraphs h1,h2, · · · ,hh of g is a mixed cycle-decomposition of g if every subgraph hi is isomorphic to some cycle ck, for k ≥ ...

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

Let F be a field, G = (V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G) consists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F, G). Then mr(F, G...

A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on n vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most O(n/ logn) times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for...

Let F be a field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F,G) consists of the symmetric irreducible tridiagonal matrices. Let mr(F,G) be the minimum rank over all matrices in S(F,G). Then mr(F,G) = 1...

let g=(v,e) be a simple graph. an edge labeling f:e to {0,1} induces a vertex labeling f^+:v to z_2 defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$, where z_2={0,1} is the additive group of order 2. for $iin{0,1}$, let e_f(i)=|f^{-1}(i)| and v_f(i)=|(f^+)^{-1}(i)|. a labeling f is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. i_f(g)=v_f(1)-v_f(0) is called the edge-f...