We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Chromatic Sum Problem is NP-complete on planar bipartite graphs with ∆ ≤ 5, but polynomial on bipartite graphs with ∆ ≤ 3, for which we construct an O(n)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bip...

A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, are assigned different colors. The strong chromatic index of G is the smallest number k for which G has a strong k-edge-coloring. A Halin graph is a planar graph consisting of a tree with no vertex of degree tw...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

Visual masking typically occurs when mask and target are separated in time by less than 100 ms, and the form of this interaction might be expected to depend on the latency of the target and mask signals. We track psychophysically the time course of signals from the two colouropponent channels by using forward and backward masking, in which mask and target each stimulate only one colour channel....

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G is the least number of colors in an acyclic edge coloring of G. In this paper, it is proved that the acyclic edge chromatic number of a planar graph G is at most ∆(G)+2 if G contains no i-cycles, 4≤ i≤ 8, or any two 3-cycles are not incident with a common vertex and ...