Murong Xu

The University of Scranton · Department of Mathematics

A (k,r)-coloring of a graph G is a proper k-vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min{d,r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k,r)-coloring. This article is intended to survey the recent developments on the stud...

Boesch and McHugh in [J. Combinatorial Theory Ser. B 38 (1985), 1-7] introduced the edge-maximal \((k, \ell )\)-graphs to study of network subcohesion, and obtained best possible upper size bounds for all edge-maximal \((k, \ell )\)-graphs. The best possible lower bounds are obtained in [J. Graph Theory 18 (1994), 227-240]. Let \(k,\ell > 0\) be in...

A (k, r)-coloring of a graph G is a proper k-vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min{d, r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k, r)-coloring. Let f(r)=r+3 if 1 ≤ r ≤ 2, f(r)=r+5 if 3 ≤ r ≤ 7 and f(r)=⌊3r/2⌋+1...

Let H = (V, E) be a hypergraph, where V is a set of vertices and E is a set of non-empty subsets of V called edges. If all edges of H have the same cardinality r, then H is an r-uniform hypergraph; if E consists of all r-subsets of V, then H is a complete r-uniform hypergraph, denoted by Krn, where n = |V |. An r-uniform hypergraph H = (V, E) is (k...

For integers k,r>0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v),r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k,r)-coloring. We prove the followin...

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$. A $r$-uniform h...

An r-hued coloring is a proper coloring such that the number of colors used by the neighbors of v is at least min{r,d(v)}. A linear r-hued coloring is an r-hued coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list r-hued chromatic number, denoted by χL,rℓ(G), of sparse graphs. It is clear that an...

For a digraph D, let $\lambda (D)$ be the arc-strong-connectivity of D. For an integer $k > 0$, a simple digraph D with $|V(D)| \ge k+1$ is k-maximal if every subdigraph H of D satisfies $\lambda (H) \le k$ but for adding new arc to D results in a subdigraph $H'$ with $\lambda (H') \ge k + 1$. We prove that if D is a simple k-maximal digraph on $n...

For a digraph D, let lambda(D) be the arc-strong-connectivity of D. For an integer k > 0, a simple digraph D with vertical bar V (D)vertical bar >= k + 1 is k-maximal if every subdigraph H of D satisfies lambda(H) <= k but for adding new arc to D results in a subdigraph H' with lambda(H') >= k + 1. We prove that if D is a simple k-maximal digraph o...

Let be an integer and let D be a simple digraph on vertices. We prove that If then D must have a nontrivial subdigraph H such that the strong arc connectivity of H is at least . We also show that this bound is best possible and present a constructive characterization for the extremal graphs.