Injective edge-coloring of subcubic graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

Planar graphs without intersecting 5-cycles are signed-4-choosable

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].

Total Face Irregularity Strength of Type (Alpha, Beta, Gamma) of Grid Graphs

We investigate new graph characteristics namely total (vertex, edge) face irregularity strength of gen- eralized plane grid graphs Gmn under k-labeling Phi of type (Alpha, Beta, Gamma). The minimum integer k for which a vertex-edge labelled graph has distinct face weights is called the total face irregularity strength of the graph and is denoted by tfs(Gmn). In this article, the graphs G = (V;E; F) under consideration are simple, finite, undirected and planar. We will estimate the exact tight lower bounds for the total face irregularity strength of some families of generalized plane grid graphs.

Total Face Irregularity Strength of Type (Alpha, Beta, Gamma) of Grid Graphs

We investigate new graph characteristics namely total (vertex, edge) face irregularity strength of gen- eralized plane grid graphs Gmn under k-labeling Phi of type (Alpha, Beta, Gamma). The minimum integer k for which a vertex-edge labelled graph has distinct face weights is called the total face irregularity strength of the graph and is denoted by tfs(Gmn). In this article, the graphs G = (V;E; F) under consideration are simple, finite, undirected and planar. We will estimate the exact tight lower bounds for the total face irregularity strength of some families of generalized plane grid graphs.

Exact Minimum Codegree Thresholds for $K_4^-$-Covering and $K_5^-$-Covering

Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let $c_2(n,F)$ be the minimum integer $t$ such that every 3-graph with minimum codegree greater than $t$ has an $F$-covering. In this note, we answer an open problem of Falgas-Ravry and Zhao (SIAM J. Discrete Math., 2016) by determining the exact value of $c_2(n, K_4^-)$ and $c_2(n, K_5^-)$, where $K_t^-$ is the complete $3$-graph on $t$ vertices with one edge removed.

Proper Orientations of Chordal Graphs

An orientation D of a graph G = (V, E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v ∈ V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u) ≠ dD−(v), for all uv ∈ E(G). An orientation with maximum indegree at most k is called a k-orientation. The proper orientation number of G, denoted by χ→(G), is the minimum integer k such that G admits a proper k-orientation. We prove that determining whether χ→(G) ≤ k is NP-complete for chordal graphs of bounded diameter. We also present a tight upper bound for χ→(G) on split graphs and a linear-time algorithm for quasi-threshold graphs.

Computing total edge irregularity strength of some n-uniform cactus chain graphs and related chain graphs

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Given graph </span><em>G</em><span>(</span><span><em>V</em>,<em>E</em></span><span>)</span><span>. We use the notion of total </span><em>k</em><span>-labeling which is edge irregular. The notion </span>of total edge irregularity strength (tes) of graph <em>G</em> means the minimum integer <em>k</em> used in the edge irregular total k-labeling of <em>G</em>. A cactus graph <em>G</em> is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is cycle <em>C_n</em> with same size <em>n</em> is named an <em>n</em>-uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then <em>G</em> is called <em>n</em>-uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs <em>C</em>(<em>C_n^r</em>) of length <em>r</em> for some <em>n</em> ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs <em>T_r</em>(4,<em>n</em>) and <em>T_r</em>(5,<em>n</em>) of length <em>r</em>. Our results are as follows: tes(<em>C</em>(<em>C_n^r</em>)) = ⌈(<em>nr</em> + 2)/3⌉ ; tes(<em>T_r</em>(4,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉ ; tes(<em>T_r</em>(5,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉.</p></div></div></div>

r-Hued coloring of planar graphs without short cycles

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

Coloring 3-power of 3-subdivision of subcubic graph

Let [Formula: see text] be a graph and [Formula: see text] be a positive integer. The [Formula: see text]-subdivision [Formula: see text] of [Formula: see text] is the graph obtained from [Formula: see text] by replacing each edge by a path of length [Formula: see text]. The [Formula: see text]-power [Formula: see text] of [Formula: see text] is the graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if the distance [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] is at most [Formula: see text]. Note that [Formula: see text] is the total graph [Formula: see text] of [Formula: see text]. The chromatic number [Formula: see text] of [Formula: see text] is the minimum integer [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-coloring. The total chromatic number of [Formula: see text], denoted by [Formula: see text], is the chromatic number of [Formula: see text]. Rosenfeld [On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396–402] and independently, Vijayaditya [On total chromatic number of a graph, J. London Math. Soc. 2 (1971) 405–408] showed that for a subcubic graph [Formula: see text], [Formula: see text]. In this note, we prove that for a subcubic graph [Formula: see text], [Formula: see text].

On t-relaxed chromatic number of r-power paths

Let [Formula: see text] be a graph and [Formula: see text] a non-negative integer. Suppose [Formula: see text] is a mapping from the vertex set of [Formula: see text] to [Formula: see text]. If, for any vertex [Formula: see text] of [Formula: see text], the number of neighbors [Formula: see text] of [Formula: see text] with [Formula: see text] is less than or equal to [Formula: see text], then [Formula: see text] is called a [Formula: see text]-relaxed [Formula: see text]-coloring of [Formula: see text]. And [Formula: see text] is said to be [Formula: see text]-colorable. The [Formula: see text]-relaxed chromatic number of [Formula: see text], denote by [Formula: see text], is defined as the minimum integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-colorable. Let [Formula: see text] and [Formula: see text] be two positive integers with [Formula: see text]. Denote by [Formula: see text] the path on [Formula: see text] vertices and by [Formula: see text] the [Formula: see text]th power of [Formula: see text]. This paper determines the [Formula: see text]-relaxed chromatic number of [Formula: see text] the [Formula: see text]th power of [Formula: see text].