IC-planar Graphs Research Articles

The Linear Arboricity of IC-planar Graphs

The linear arboricity of a graph is the minimum number of linear forests that partition the edges of . In 1981, Akiyama, Exoo and Harary conjectured that for any simple graph . A graph is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. An IC-planar graph is a 1-planar graph satisfying the condition that each vertex is incident with at most one crossing edge. It is shown in this paper that every IC-planar graph with maximum degree has .

Improper odd coloring of IC-planar graphs

An IC-planar graph is a graph that can be drawn in the plane in such a way that each edge is crossed at most once and each vertex is incident with at most one crossed edge. In this paper, we show that every IC-planar graph can be colored with nine colors so that for every non-isolate vertex there exists a color occurring odd times in its neighbors.

The linear 2-arboricity of IC-planar graphs

The linear 2-arboricity of a graph G, denoted by la2(G), is the least integer k such that G can be partitioned into k edge-disjoint forests, whose components are paths of length at most 2. A 1-planar graph is a graph that can be drawn in the Euclidean plane such that each edge crosses at most one edge. An IC-planar is a 1-planar graph satisfying the condition that each vertex is incident with at most one crossing edge. It is shown in this paper that every IC-planar graph G with maximum degree Δ has la2(G)≤⌈Δ+12⌉+6.

Equitable Coloring of IC-Planar Graphs with Girth g ≥ 7

An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth g≥7 is equitably Δ(G)-colorable, where Δ(G) is the maximum degree of G.

IC-planar graphs are odd-10-colorable

An odd-k-coloring of a graph G is a proper k-coloring such that for every vertex v∈V(G) with d(v)≥1, there exists a color occurring odd times in its neighbors. The odd chromatic number, denote by χo(G), of G is the smallest integer k so that G admits an odd-k-coloring. Call a graph G IC-planar if it can be drawn in the plane so that every edge is crossed by at most once and every vertex has at most one incident edge being crossed. It was shown that every planar graph G has χo(G)≤8. In this paper we show that every IC-planar graph G satisfies χo(G)≤10.

Neighbor sum distinguishing total choice number of IC-planar graphs with restrictive conditions

<abstract><p>A neighbor sum distinguishing (NSD) total coloring $ \phi $ of $ G $ is a proper total coloring such that $ \sum_{z\in E_{G}(u)\cup\{u\}}\phi(z)\neq\sum_{z\in E_{G}(v)\cup\{v\}}\phi(z) $ for each edge $ uv\in E(G) $. Pilśniak and Woźniak asserted that each graph with a maximum degree $ \Delta $ admits an NSD total $ (\Delta+3) $-coloring in 2015. In this paper, we prove that the list version of this conjecture holds for any IC-planar graph with $ \Delta\geq10 $ but without five cycles by applying the discharging method, which improves the result of Zhang (NSD list total coloring of IC-planar graphs without five cycles).</p></abstract>

Neighbor sum distinguishing total choosability of triangle-free IC-planar graphs

Neighbor sum distinguishing total choosability of triangle-free IC-planar graphs

The Surviving Rate of IC-Planar Graphs

Let k and n be two positive integers. Firefighting is a discrete dynamical process of preventing the spread of fire. Let G be a connected graph G with n vertices. Assuming a fire starts at one of the vertices of G, the firefighters choose k unburned vertices at each step, and then the fire spreads to all unprotected neighbors of the burning vertices. The process continues until the fire stops spreading. The goal is to protect as many vertices as possible. When a fire breaks out randomly at a vertex of G, its k-surviving rate, ρk(G), is the expected number of saved vertices. A graph is IC-planar if it has a drawing in which each edge cross once and their endpoints are disjoint. In this paper, we prove that ρ4(G)>1124 for every IC-planar graph G. This is proven by the discharging method and the locally symmetric of the graph.

The Stub Resolution of 1-planar Graphs

The resolution of a drawing plays a crucial role when defining criteria for its quality. In the past, grid resolution, edge-length resolution, angular resolution and crossing resolution have been investigated. In this paper, we investigate the stub resolution, a recently introduced criterion for nonplanar drawings. Intersection points divide edges into parts, called stubs, which should not be too short for the sake of readability. Thus, the stub resolution of a drawing is defined as the minimum ratio between the length of a stub and the length of the entire edge, over all the edges of the drawing. We consider $1$-planar graphs and we explore scenarios in which near optimal stub resolution, i.e., arbitrarily close to $\frac{1}{2}$, can be obtained in drawings with zero, one or two bends per edge, as well as further resolution criteria, such as angular and crossing resolution. In particular, our main contributions are as follows: (i) Every IC-planar graph, i.e., every $1$-planar graph with independent crossing edges, has a straight-line drawing with near optimal stub resolution; (ii) Every $1$-planar graph has a $1$-bend drawing with near optimal stub resolution.

On the Total Neighbor Sum Distinguishing Index of IC-Planar Graphs

A proper total k-coloring ϕ of G with ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each uv∈E(G) is called a total neighbor sum distinguishing k-coloring, where EG(u)={uv|uv∈E(G)}. Pilśniak and Woźniak conjectured that every graph with maximum degree Δ exists a total neighbor sum distinguishing (Δ+3)-coloring. In this paper, we proved that any IC-planar graph with Δ≥12 satisfies this conjecture, which improves the result of Song and Xu.

A Sufficient Condition for an IC-Planar Graph to be Class 1

A Sufficient Condition for an IC-Planar Graph to be Class 1

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

Acyclic Edge Coloring of IC-planar Graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G is the least number of colors such that G has an acyclic edge coloring and denoted by Χ′a(G). An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. In this paper, it is proved that Χ′a(G) ≤ Δ(G) + 10, if G is an IC-planar graph without adjacent triangles and Χ′a(G) ≤ Δ(G) + 8, if G is a triangle-free IC-planar graph.

Neighbor sum distinguishing total coloring of IC-planar graphs

Two distinct crossings are independent if the end-vertices of the crossed edge are mutually different. A graph is said to be plane graph with independent crossings or IC-planar, if it can be drawn in the plane such that every two crossings are independent. A proper total-k-coloring of G is called neighbor sum distinguishing if ∑c(u)≠∑c(v) for each edge uv∈E(G), where ∑c(v) is used to denote the sum of the color of a vertex v and the colors of edges incident with v. The least number k needed for such a coloring of G, denoted by χΣ′′(G), is the neighbor sum distinguishing total chromatic number. By using the famous Combinatorial Nullstellensatz, we prove that Δ(G)+2 colors guarantee such a total coloring for any IC-planar graphs without 2-vertex incident with crossed edge and with Δ(G)≥14.

Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs

A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex. A proper total-k-coloring of G is called neighbor sum distinguishing if ∑c(u) ≠ ∑c(v) for each edge uv ∈ E(G), where ∑c(v) denote the sum of the color of a vertex v and the colors of edges incident with v. The least number k needed for such a total coloring of G, denoted by X′′∑(G), is the neighbor sum distinguishing total chromatic number. Pilśniak and Woźniak conjectured X′′∑(G) ≤ △(G) + 3 for any simple graph with maximum degree △(G). By using the famous Combinatorial Nullstellensatz, we prove that above conjecture holds for any triangle free IC-planar graph with △(G) ≥ 7. Moreover, it holds for any triangle free planar graph with △(G) ≥ 6.

Neighbor sum distinguishing total coloring of IC-planar graphs with short cycle restrictions

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring ϕ : V(G)∪E(G)→{1,2,…,k} is neighbor sum distinguishing if fϕ(u)≠fϕ(v) for each uv∈E(G), where fϕ(v)=∑uv∈E(G)ϕ(uv)+ϕ(v), v∈V(G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ′′(G). In this paper, by using the discharging method, we prove that χΣ′′(G)≤max{Δ(G)+3,10} if G is a triangle free IC-planar graph and χΣ′′(G)≤max{Δ(G)+3,13} if G is an IC-planar graph without adjacent triangles, where Δ(G) is the maximum degree of G.

不含4-围的IC-平面图的线性荫度

不含4-围的IC-平面图的线性荫度

Neighbor sum distinguishing total choosability of IC-planar graphs

Neighbor sum distinguishing total choosability of IC-planar graphs

Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13

A graph is IC-planar if it admits a drawing on the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring $$\phi $$ : $$V(G)~\cup ~E(G)\rightarrow \{1,2,\ldots ,k\}$$ is called neighbor sum distinguishing if $$f_{\phi }(u)\ne f_{\phi }(v)$$ for each $$uv\in E(G)$$, where $$f_{\phi }(u)$$ is the sum of the color of u and the colors of the edges incident with u. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by $$\chi ''_{\Sigma }(G)$$. Pilśniak and Woźniak conjectured $$\chi _{\Sigma }''(G)\le \Delta (G)+3$$ for any simple graph with maximum degree $$\Delta (G)$$. This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph G with $$\Delta (G)=13$$.

Acyclic coloring of IC-planar graphs

A graph G is called an IC-planar graph if it can be embedded in the plane so that every edge is crossed by at most one other edge and every vertex is incident to at most one crossing edge. In this paper, we prove that every IC-planar graph is acyclically 10-colorable. Moreover, an IC-planar graph of the acyclic chromatic number 6 is constructed.