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Register a new userNote on the group edge irregularity strength of graphs
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We investigate the textit{edge group irregularity strength} ($es_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $mathcal{G}$ of order $s$, there exists a function $f:V(G)rightarrow mathcal{G}$ such that the sums of vertex labels at every edge are distinct. In this note we provide some upper bounds on $es_g(G)$ as well as for edge irregularity strength $es(G)$ and harmonious order $rm{har}(G)$.
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We investigate the textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $gr$ of order $s$, there exists a function $f:E(G)rightarrow gr$ such that the sums of edge labels at every vertex are distinct. So far it was not known if $s_g(G)$ is bounded for disconnected graphs. In the paper we we present some upper bound for all graphs. Moreover we give the exact values and bounds on $s_g(G)$ for disconnected graphs without a star as a component.
We investigate the textit{group irregularity strength}, $s_g(G)$, of a graph, i.e. the least integer $k$ such that taking any Abelian group $mathcal{G}$ of order $k$, there exists a function $f:E(G)rightarrow mathcal{G}$ so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on $s_g(G)$ for a general graph $G$ was exponential in $n-c$, where $n$ is the order of $G$ and $c$ denotes the number of its components. In this note we prove that $s_g(G)$ is linear in $n$, namely not greater than $2n$. In fact, we prove a stronger result, as we additionally forbid the identity element of a group to be an edge label or the sum of labels around a vertex. We consider also locally irregular labelings where we require only sums of adjacent vertices to be distinct. For the corresponding graph invariant we prove the general upper bound: $Delta(G)+{rm col}(G)-1$ (where ${rm col}(G)$ is the coloring number of $G$) in the case when we do not use the identity element as an edge label, and a slightly worse one if we additionally forbid it as the sum of labels around a vertex. In the both cases we also provide a sharp upper bound for trees and a constant upper bound for the family of planar graphs.
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $delta^c(G)>frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $delta^c(G)>frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $ngeq 8k-18$ and $delta^c(G)>frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.
A semi-proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: A(D)rightarrow mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denoted by $w^-_D(v)$, is the sum of the weights of arcs towards $v$. The semi-proper orientation number of a graph $G$, denoted by $overrightarrow{chi}_s(G)$, is the minimum of maximum in-weight of $v$ in $D$ over all semi-proper orientation $(D,w)$ of $G$. This parameter was first introduced by Dehghan (2019). When the weights of all edges eqaul to one, this parameter is equal to the proper orientation number of $G$. The optimal semi-proper orientation is a semi-proper orientation $(D,w)$ such that $max_{vin V(G)}w_D^-(v)=overrightarrow{chi}_s(G)$. Araujo et al. (2016) showed that $overrightarrow{chi}(G)le 7$ for every cactus $G$ and the bound is tight. We prove that for every cactus $G$, $overrightarrow{chi}_s(G) le 3$ and the bound is tight. Ara{u}jo et al. (2015) asked whether there is a constant $c$ such that $overrightarrow{chi}(G)le c$ for all outerplanar graphs $G.$ While this problem remains open, we consider it in the weighted case. We prove that for every outerplanar graph $G,$ $overrightarrow{chi}_s(G)le 4$ and the bound is tight.
Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $chi(G)$, and the maximum is the so-called Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let $chi_c(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $chi_c(G)>chi(G)$. We prove that $chi(G)=chi_c(G)$ if $G$ is bipartite, and that $chi_c(G)leq 4$ if $G$ is subcubic.
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