Properties of minimally t-tough graphs

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta \left(G\right)=2$. We show that in every minimally $1$-tough graph $\delta \left(G\right)\le \frac{n}{3}+1$. We also prove that every minimally $1$-tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number $t$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph.     

Problems on matchings and independent sets of a graph

Let $G$ be a finite simple graph. For $X?V\left(G\right)$, the difference of $X$, $d\left(X\right)?\left|X\right|?\left|N\left(X\right)\right|$ where $N\left(X\right)$ is the neighborhood of $X$ and $max\{d\left(X\right):X?V\left(G\right)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d\left(X\right)$ equals the critical difference and $\ker \left(G\right)$ is the intersection of all critical sets. $\mathrm{diadem}\left(G\right)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with$d\left(S\right)>0$ if no proper subset of $S$ has positive difference.A graph $G$ is called a König–Egerváry graph if the sum of its independence number $\alpha \left(G\right)$ and matching number $\mu \left(G\right)$ equals $\left|V\left(G\right)\right|$.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set $S$ with $d\left(S\right)>0$ is at least the critical difference of the graph.We also give a new short proof of the inequality $\left|\ker \left(G\right)\right|+\left|\mathrm{diadem}\left(G\right)\right|\le 2\alpha \left(G\right)$.A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha \left(G\right)?\mu \left(G\right)$, is proved.We also make an observation about $\ker \left(G\right)$ using Edmonds–Gallai Structure Theorem as a concluding remark.     

Star saturation number of random graphs

For a given graph $F$, the $F$-saturation number of a graph $G$ is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. Recently, the $F$-saturation number of the Erd?s–Rényi random graph $\mathbb{G}(n,p)$ has been determined asymptotically for any complete graph $F$. In this paper, we give an asymptotic formula for the $F$-saturation number of $\mathbb{G}(n,p)$ when $F$ is a star graph.     

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.     

Reconfiguration graphs of shortest paths

For a graph $G$ and$a,b\in V\left(G\right)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and$b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and$b$ in $G$. Two vertices in $V\left(S(G,a,b)\right)$ are adjacent, if their corresponding paths in $G$ differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice.     

A note on the list vertex arboricity of toroidal graphs

The vertex arboricity $a\left(G\right)$ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ such that no cycle is monochromatic. The list vertex arboricity $a_l\left(G\right)$ is the list-coloring version of this concept. In this note, we prove that if $G$ is a toroidal graph, then $a_l\left(G\right)\le 4$; and $a_l\left(G\right)=4$ if and only if $G$ contains $K_7$ as an induced subgraph.     

Histories in Path Graphs

For a given graph G and a positive integer r the r-path graph, $P_r(G)$, has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length $r-1$, and their union forms either a cycle or a path of length $r+1$ in G. Let $P_r^k(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_r^k(G)$. The k-history $P_r^{-k}(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.     

Compatible Eulerian circuits in Eulerian (di)graphs with generalized transition systems

A transition in a graph is defined as a pair of adjacent edges. A transition system of an Eulerian graph refers to a set of partitions such that for each vertex of the graph, there corresponds to a partition of the set of edges incident to the vertex into transitions. A generalized transition system $F\left(G\right)$ over a graph $G$ defines a set of transitions over $G$. A compatible Eulerian circuit of an Eulerian graph $G$ with a generalized transition system $F\left(G\right)$ is defined as an Eulerian circuit in which no two consecutive edges form a transition defined by $F\left(G\right)$. In this paper, we further introduce the concept of weakly generalized transition system which is an extension of the generalized transition system and prove some Ore-type sufficient conditions for the existence of compatible Eulerian circuits in Eulerian graphs with (weakly) generalized transition systems and obtain corresponding results for Eulerian digraphs. Our conditions improve some previous results due to Jackson and Isaak, respectively.     

Pentavalent symmetric graphs admitting transitive non-abelian characteristically simple groups

Let $\Gamma$ be a graph and let $G$ be a group of automorphisms of $\Gamma$. The graph $\Gamma$ is called $G$-normal if $G$ is normal in the automorphism group of $\Gamma$. Let $T$ be a finite non-abelian simple group and let $G=T^l$ with $l\ge 1$. In this paper we prove that if every connected pentavalent symmetric $T$-vertex-transitive graph is $T$-normal, then every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal. This result, among others, implies that every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal except $T$ is one of 57 simple groups. Furthermore, every connected pentavalent symmetric $G$-regular graph is $G$-normal except $T$ is one of 20 simple groups, and every connected pentavalent $G$-symmetric graph is $G$-normal except $T$ is one of 17 simple groups.     

A sufficient condition for Kim’s conjecture on the competition numbers of graphs

A hole of a graph $G$ is an induced cycle of length at least 4. Kim (2005) [3] conjectured that the competition number $k\left(G\right)$ is bounded by $h\left(G\right)+1$ for any graph $G$, where $h\left(G\right)$ is the number of holes of $G$. Li and Chang (2009) [5] proved that the conjecture is true for a graph whose holes all satisfy a property called ‘independence’. In this paper, by using similar proof techniques in Li and Chang (2009) [5], we prove the conjecture for graphs satisfying two conditions that allow the holes to overlap a lot.     

Injective choosability of subcubic planar graphs with girth 6

An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $\left|L\left(v\right)\right|\ge k$ for all $v\in V\left(G\right)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens the result of Lu?ar, ?krekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases.     

The strong chromatic index of Halin graphs

A strong edge coloring of a graph $G$ is an assignment of colors to the edges of $G$ such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index of a graph $G$, denoted by $s{\chi }^{\prime }\left(G\right)$, is the minimum number of colors needed for a strong edge coloring of $G$. A Halin graph $G$ is a plane graph constructed from a tree $T$ without vertices of degree two by connecting all leaves through a cycle $C$. If a Halin graph $G=T\cup C$ is different from a certain necklace $N{e}_2$ and any wheel $W_n$, $n?0(mod3)$, then we prove that $s{\chi }^{\prime }\left(G\right)?s{\chi }^{\prime }\left(T\right)+3$.     

A panconnectivity theorem for bipartite graphs

Let $G$ be a simple $m\times n$ bipartite graph with $m\ge n$. We prove that if the minimum degree of $G$ satisfies $\delta \left(G\right)\ge m∕2+1$, then $G$ is bipanconnected: for every pair of vertices $x,y$, and for every appropriate integer $2\le ?\le 2n$, there is an $x,y$-path of length $?$ in $G$.