Doppelgangers and Lemke graphs

Let $G$ be a connected graph. A configuration of pebbles on $G$ is a function that assigns a nonnegative integer to each vertex. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A configuration is solvable if after making pebbling moves any vertex can get at least one pebble. The pebbling number of $G$, denoted $\pi \left(G\right)$, is the smallest integer such that any configuration of $\pi \left(G\right)$ pebbles on $G$ is solvable. A graph has the two-pebbling property if after placing more than $2\pi \left(G\right)?q$ pebbles on $G$, where $q$ is the number of vertices with pebbles, there is a sequence of pebbling moves so that at least two pebbles can be placed on any vertex. A graph without the two-pebbling property is called a Lemke graph. Previously, an infinite family of Lemke graphs was shown to exist by subdividing edges of the original Lemke graph. In this paper, we introduce a new way to create infinite families of Lemke graphs based on adding vertices as well as subdividing edges. We also characterize the configurations that violate the two-pebbling property on these graphs and conjecture another infinite family of Lemke graphs that generalizes the original Lemke graph.     

A local condition for k-contractible edges

An edge of a $k$-connected graph is said to be $k$-contractible if the contraction of the edge results in a $k$-connected graph. For a graph $G$ and a vertex $x$ of $G$, let $G\left[N_G\left(x\right)\right]$ be the subgraph induced by the neighborhood of $x$. We prove that if $G\left[N_G\left(x\right)\right]$ has less than $?\frac{k}{2}?$ edges for any vertex $x$ of a $k$-connected graph $G$, then $G$ has a $k$-contractible edge. We also show that the bound $?\frac{k}{2}?$ is sharp.     

Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles

For any graph $G$ with $a,b\in V\left(G\right)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V\left(S(G,a,b)\right)$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles.     

On s-hamiltonian line graphs of claw-free graphs

For an integer $s\ge 0$, a graph $G$ is $s$-hamiltonian if for any vertex subset $S?V\left(G\right)$ with $\left|S\right|\le s$, $G?S$ is hamiltonian, and $G$ is $s$-hamiltonian connected if for any vertex subset $S?V\left(G\right)$ with $\left|S\right|\le s$, $G?S$ is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Ku?zel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjá?ek and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of $s$-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for $s\ge 2$, a line graph $L\left(G\right)$ is $s$-hamiltonian if and only if $L\left(G\right)$ is $(s+2)$-connected. In this paper we prove the following.(i) For an integer $s\ge 2$, the line graph $L\left(G\right)$ of a claw-free graph $G$ is $s$-hamiltonian if and only if $L\left(G\right)$ is $(s+2)$-connected.(ii) The line graph $L\left(G\right)$ of a claw-free graph $G$ is 1-hamiltonian connected if and only if $L\left(G\right)$ is 4-connected.     

A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

Relation between the H-rank of a mixed graph and the rank of its underlying graph

Given a simple graph $G=(V_G,E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\stackrel{?}{G}$ is obtained from $G$ by orienting some of its edges. Let $H\left(\stackrel{?}{G}\right)$ denote the Hermitian adjacency matrix of $\stackrel{?}{G}$ and $A\left(G\right)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\stackrel{?}{G}$ (resp. $G$), written as $rk\left(\stackrel{?}{G}\right)$ (resp. $r\left(G\right)$), is the rank of $H\left(\stackrel{?}{G}\right)$ (resp. $A\left(G\right)$). Denote by $d\left(G\right)$ the dimension of cycle space of $G$, that is $d\left(G\right)=\left|E_G\right|?\left|V_G\right|+\omega \left(G\right)$, where $\omega \left(G\right)$ denotes the number of connected components of $G$. In this paper, we concentrate on the relation between the $H$-rank of $\stackrel{?}{G}$ and the rank of $G$. We first show that $?2d\left(G\right)?rk\left(\stackrel{?}{G}\right)?r\left(G\right)?2d\left(G\right)$ for every mixed graph $\stackrel{?}{G}$. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in Luo et al. (2018); Wong et al. (2016) may be deduced consequently.     

On the powers of vertex cover ideals

Let $S=\mathbb{K}[x_1,\dots ,x_n]$ be a polynomial ring, where $\mathbb{K}$ is a field, and G be a simple graph on n vertices. Let $J(G)?S$ be the vertex cover ideal of G. Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if G is a unicyclic vertex decomposable graph, then symbolic powers of $J(G)$ are componentwise linear.     

Proportional choosability: A new list analogue of equitable coloring

In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A $k$-assignment $L$ for a graph $G$ specifies a list $L\left(v\right)$ of $k$ available colors for each vertex $v$ of $G$. An $L$-coloring assigns a color to each vertex $v$ from its list $L\left(v\right)$. For each color $c$, let $\eta \left(c\right)$ be the number of vertices $v$ whose list $L\left(v\right)$ contains $c$. A proportional$L$-coloring of $G$ is a proper $L$-coloring in which each color $c\in {?}_{v\in V\left(G\right)}L\left(v\right)$ is used $?\eta \left(c\right)∕k?$ or $?\eta \left(c\right)∕k?$ times. A graph $G$ is proportionally$k$-choosable if a proportional $L$-coloring of $G$ exists whenever $L$ is a $k$-assignment for $G$. We show that if a graph $G$ is proportionally $k$-choosable, then every subgraph of $G$ is also proportionally $k$-choosable and also $G$ is proportionally $(k+1)$-choosable, unlike equitable choosability for which analogous claims would be false. We also show that any graph $G$ is proportionally $k$-choosable whenever $k\ge \Delta \left(G\right)+?\left|V\left(G\right)\right|∕2?$, and we use matching theory to completely characterize the proportional choosability of stars and the disjoint union of cliques.     

A tight bound for conflict-free coloring in terms of distance to cluster

Given an undirected graph $G=(V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of G, denoted by ${\chi }_{ON}(G)$.In previous work [WG 2020], we showed the upper bound ${\chi }_{ON}(G)\le \mathsf{dc}(G)+3$, where $\mathsf{dc}(G)$ denotes the distance to cluster parameter of G. In this paper, we obtain the improved upper bound of ${\chi }_{ON}(G)\le \mathsf{dc}(G)+1$. We also exhibit a family of graphs for which ${\chi }_{ON}(G)>\mathsf{dc}(G)$, thereby demonstrating that our upper bound is tight.     

Spectra of strongly Deza graphs

A Deza graph G with parameters $(n,k,b,a)$ is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children $G_A$ and $G_B$ of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in $G_A$ or $G_B$ if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular.     

Domination versus total domination in claw-free cubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number $\gamma (G)$ of G is the minimum cardinality of a dominating set in G, while the total domination number ${\gamma }_t(G)$ of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then $\frac{{\gamma }_t(G)}{\gamma (G)}\le \frac{12}{7}$, and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then ${\gamma }_t(G)\le \frac{3}{7}n$, and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19].     

Relating broadcast independence and independence

An independent broadcast on a connected graph $G$ is a function $f:V\left(G\right)\to {\mathbb{N}}_0$ such that, for every vertex $x$ of $G$, the value $f\left(x\right)$ is at most the eccentricity of $x$ in $G$, and $f\left(x\right)>0$ implies that $f\left(y\right)=0$ for every vertex $y$ of $G$ within distance at most $f\left(x\right)$ from $x$. The broadcast independence number ${\alpha }_b\left(G\right)$ of $G$ is the largest weight $\sum _{x\in V\left(G\right)}f\left(x\right)$ of an independent broadcast $f$ on $G$. Clearly, ${\alpha }_b\left(G\right)$ is at least the independence number $\alpha \left(G\right)$ for every connected graph $G$. Our main result implies ${\alpha }_b\left(G\right)\le 4\alpha \left(G\right)$. We prove a tight inequality and characterize all extremal graphs.     

Connected k-dominating graphs

For a graph $G=(V,E)$, the $k$-dominating graph $D_k\left(G\right)$ of $G$ has vertices corresponding to the dominating sets of $G$ having cardinality at most $k$, where two vertices of $D_k\left(G\right)$ are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote the domination and upper domination numbers of $G$ by $\gamma \left(G\right)$ and $\Gamma \left(G\right)$, respectively, and the smallest integer $\epsilon$ for which $D_k\left(G\right)$ is connected for all $k\ge \epsilon$ by $d_0\left(G\right)$. It is known that $\Gamma \left(G\right)+1\le d_0\left(G\right)\le \left|V\right|$, but constructing a graph $G$ such that $d_0\left(G\right)>\Gamma \left(G\right)+1$ appears to be difficult.We present two related constructions. The first construction shows that for each integer $k\ge 3$ and each integer $r$ such that $1\le r\le k?1$, there exists a graph $G_{k,r}$ such that $\Gamma \left(G_{k,r}\right)=k$, $\gamma \left(G_{k,r}\right)=r+1$ and $d_0\left(G_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)?1$. The second construction shows that for each integer $k\ge 3$ and each integer $r$ such that $1\le r\le k?1$, there exists a graph $Q_{k,r}$ such that $\Gamma \left(Q_{k,r}\right)=k$, $\gamma \left(Q_{k,r}\right)=r$ and $d_0\left(Q_{k,r}\right)=k+r=\Gamma \left(G\right)+\gamma \left(G\right)$.     

On the (3,1)-choosability of planar graphs without adjacent cycles of length 5,6,7

A $(k,d)$-list assignment $L$ of a graph $G$ is a mapping that assigns to each vertex $v$ a list $L\left(v\right)$ of at least $k$ colors satisfying $|L\left(x\right)\cap L\left(y\right)|\le d$ for each edge $xy$. A graph $G$ is $(k,d)$-choosable if there exists an $L$-coloring of $G$ for every $(k,d)$-list assignment $L$. This concept is also known as choosability with separation. In this paper, we prove that any planar graph $G$ is $(3,1)$-choosable if any $i$-cycle is not adjacent to a $j$-cycle, where $5\le i\le 6$ and $5\le j\le 7$.     

An improvement to the Hilton-Zhao vertex-splitting conjecture

For a simple graph G, denote by n, $\mathrm{\Delta }(G)$, and ${\chi }^{\prime }(G)$ its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)<{\chi }^{\prime }(G)$ for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let $G^{?}$ be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that $G^{?}$ must be edge-chromatic critical if $\mathrm{\Delta }(G)>n/3$, and they verified this when $\mathrm{\Delta }(G)\ge \frac{n}{2}(\sqrt{7}?1)\approx 0.82n$. In this paper, we prove it for $\mathrm{\Delta }(G)\ge 0.75n$.     

On the signless Laplacian spectral radius of weighted digraphs

Let $G=(V\left(G\right),E\left(G\right))$ be a weighted digraph with vertex set $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and arc set $E\left(G\right)$, where the arc weights are nonzero nonnegative symmetric matrices. In this paper, we obtain an upper bound on the signless Laplacian spectral radius of a weighted digraph $G$, and if $G$ is strongly connected, we also characterize the digraphs achieving the upper bound. Moreover, we show that an upper bound of weighted digraphs or unweighted digraphs can be deduced from our upper bound.