Equitable Δ-coloring of graphs

Consider a graph $G$ consisting of a vertex set $V\left(G\right)$ and an edge set $E\left(G\right)$. Let $\Delta \left(G\right)$ and $\chi \left(G\right)$ denote the maximum degree and the chromatic number of $G$, respectively. We say that $G$ is equitably $\Delta \left(G\right)$-colorable if there exists a proper $\Delta \left(G\right)$-coloring of $G$ such that the sizes of any two color classes differ by at most one. Obviously, if $G$ is equitably $\Delta \left(G\right)$-colorable, then $\Delta \left(G\right)\ge \chi \left(G\right)$. Conversely, even if $G$ satisfies $\Delta \left(G\right)\ge \chi \left(G\right)$, we cannot guarantee that $G$ must be equitably $\Delta \left(G\right)$-colorable. In 1994, the Equitable $\Delta$-Coloring Conjecture $\left(E\Delta \mathrm{CC}\right)$ asserts that a connected graph $G$ with $\Delta \left(G\right)\ge \chi \left(G\right)$ is equitably $\Delta \left(G\right)$-colorable if $G$ is different from $K_{2n+1,2n+1}$ for all $n\ge 1$. In this paper, we give necessary conditions for a graph $G$ (not necessarily connected) with $\Delta \left(G\right)\ge \chi \left(G\right)$ to be equitably $\Delta \left(G\right)$-colorable and prove that those necessary conditions are also sufficient conditions when $G$ is a bipartite graph, or $G$ satisfies $\Delta \left(G\right)\ge \frac{\left|V\left(G\right)\right|}{3}+1$, or $G$ satisfies $\Delta \left(G\right)\le 3$.     

Homomorphism complexes and k-cores

For any fixed graph $G$, we prove that the topological connectivity of the graph homomorphism complex Hom($G,K_m$) is at least $m?D\left(G\right)?2$, where $D\left(G\right)={max}_{H?G}\delta \left(H\right)$, for $\delta \left(H\right)$ the minimum degree of a vertex in a subgraph $H$. This generalizes a theorem of C?uki? and Kozlov, in which the maximum degree $\Delta \left(G\right)$ was used in place of $D\left(G\right)$, and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, $\chi \left(G\right)\le D\left(G\right)+1$, as $\chi \left(G\right)=min\{m:\text{Hom}(G,K_m)\ne ?\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c∕n$ for a fixed constant $c>0$.     

Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of a graph $G$, respectively. The matrix $D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The spectrum of the matrix $D(G)+A(G)$ is called the Q-spectrum of $G$. A graph is said to be determined by its Q-spectrum if there is no other non-isomorphic graph with the same Q-spectrum. In this paper, we prove that all starlike trees whose maximum degree exceed $4$ are determined by their Q-spectra.     

Kirchhoff index in line,subdivision and total graphs of a regular graph

Let $G$ be a connected regular graph and $l\left(G\right)$, $s\left(G\right)$, $t\left(G\right)$ the line, subdivision, total graphs of $G$, respectively. In this paper, we derive formulae and lower bounds of the Kirchhoff index of $l\left(G\right)$, $s\left(G\right)$ and $t\left(G\right)$, respectively. In particular, we give special formulae for the Kirchhoff index of $l\left(G\right)$, $s\left(G\right)$ and $t\left(G\right)$, where $G$ is a complete graph $K_n$, a regular complete bipartite graph $K_{n,n}$ and a cycle $C_n$.     

An improved upper bound on the adjacent vertex distinguishing total chromatic number of graphs

An adjacent vertex distinguishing total $k$-coloring of a graph $G$ is a proper total $k$-coloring of $G$ such that any pair of adjacent vertices have different sets of colors. The minimum number $k$ needed for such a total coloring of $G$ is denoted by ${\chi }_a^{\prime \prime }\left(G\right)$. In this paper we prove that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)?1$ if $\Delta \left(G\right)\ge 4$, and ${\chi }_a^{\prime \prime }\left(G\right)\le ?\frac{5\Delta \left(G\right)+8}{3}?$ in general. This improves a result in Huang et al. (2012) which states that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)$ for any graph with $\Delta \left(G\right)\ge 3$.     

Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a mulitigraph $G$, denoted ${\chi }_s^{\prime }\left(G\right)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star$k$-edge-colorable if ${\chi }_s^{\prime }\left(G\right)\le k$. Dvo?ák et al. (2013) proved that every subcubic multigraph is star 7-edge-colorable, and conjectured that every subcubic multigraph should be star 6-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7∕3$ is star list-5-edge-colorable. It is known that a graph with maximum average degree $14∕5$ is not necessarily star 5-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12∕5$ is star 5-edge-colorable.     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).     

On maximum matchings in König–Egerváry graphs

For a graph $G$ let $\alpha \left(G\right),\mu \left(G\right)$, and $\tau \left(G\right)$ denote its independence number, matching number, and vertex cover number, respectively. If $\alpha \left(G\right)+\mu \left(G\right)=\left|V\left(G\right)\right|$ or, equivalently, $\mu \left(G\right)=\tau \left(G\right)$, then $G$ is a König–Egerváry graph.In this paper we give a new characterization of König–Egerváry graphs.     

Strong list-chromatic index of subcubic graphs

A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by ${\chi }_s^{\prime }\left(G\right)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erd?s and Ne?et?il conjectured that ${\chi }_s^{\prime }\left(G\right)\le \frac{5}{4}\Delta {\left(G\right)}^2$, where $\Delta \left(G\right)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.     

Which cospectral graphs have same degree sequences

Two graphs are said to be $L$-cospectral (respectively, $Q$-cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph $G$ is said to be $L?DS$ (respectively, $Q?DS$) if there does not exist other non-isomorphic graph $H$ such that $H$ and $G$ are $L$-cospectral (respectively, $Q$-cospectral). Let $d_1\left(G\right)\ge d_2\left(G\right)\ge ?\ge d_n\left(G\right)$ be the degree sequence of a graph $G$ with $n$ vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with $d_1\left(G\right)\in \{4,5\}$), if $H$ is $L$-cospectral (respectively, $Q$-cospectral) with a connected graph $G$ and $d_2\left(G\right)=2$, then $H$ has the same degree sequence as $G$. A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both $L?DS$, which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014).     

A note on the shameful conjecture

Let $P_G\left(q\right)$ denote the chromatic polynomial of a graph $G$ on $n$ vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, $\frac{P_G\left(n\right)}{P_G(n-1)}\ge \frac{n^n}{{(n-1)}^n}\text{.}$ Let $\mu \left(G\right)$ denote the expected number of colors used in a uniformly random proper $n$-coloring of $G$. The above inequality can be interpreted as saying that $\mu \left(G\right)\ge \mu \left(O_n\right)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F.M. Dong, who in fact showed that, $\frac{P_G\left(q\right)}{P_G(q-1)}\ge \frac{q^n}{{(q-1)}^n}$ for all $q\ge n$. There are examples showing that this inequality is not true for all $q\ge 2$. In this paper, we show that the above inequality holds for all $q\ge 36D^{3/2}$, where $D$ is the largest degree of $G$. It is also shown that the above inequality holds true for all $q\ge 2$ when $G$ is a claw-free graph.     

On the edge dimension of a graph

Given a connected graph $G(V,E)$, the edge dimension, denoted $\mathrm{edim}\left(G\right)$, is the least size of a set $S?V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise different tuples of distances to the vertices of $S$. The notation was introduced by Kelenc, Tratnik, and Yero, and in their paper they posed several questions about various properties of $\mathrm{edim}$. In this article we answer two of these questions: we classify the graphs on $n$ vertices for which $\mathrm{edim}\left(G\right)=n?1$ and show that $\frac{\mathrm{edim}\left(G\right)}{dim\left(G\right)}$ is not bounded from above (here $dim\left(G\right)$ is the standard metric dimension of $G$). We also compute $\mathrm{edim}(G\square P_m)$ and $\mathrm{edim}(G+K_1)$.     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.     

Edge-partitions of graphs and their neighbor-distinguishing index

A proper edge coloring is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The minimum number of colors needed for a neighbor-distinguishing edge coloring is the neighbor-distinguishing index, denoted by ${\chi }_a^{\prime }\left(G\right)$. A graph is normal if it contains no isolated edges. Let $G$ be a normal graph, and let $\Delta \left(G\right)$ and ${\chi }^{\prime }\left(G\right)$ denote the maximum degree and the chromatic index of $G$, respectively. We modify the previously known techniques of edge-partitioning to prove that ${\chi }_a^{\prime }\left(G\right)\le 2{\chi }^{\prime }\left(G\right)$, which implies that ${\chi }_a^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$. This improves the result in Wang et al. (2015), which states that ${\chi }_a^{\prime }\left(G\right)\le \frac{5}{2}\Delta \left(G\right)$ for any normal graph. We also prove that ${\chi }_a^{\prime }\left(G\right)\le 2\Delta \left(G\right)$ when $\Delta \left(G\right)=2^k$, $k$ is an integer with $k\ge 2$.     

A new formula for the decycling number of regular graphs

The decycling number $?\left(G\right)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resultant graph contains no cycle. A decycling set containing exactly $?\left(G\right)$ vertices of $G$ is called a $?$-set. For any decycling set $S$ of a $k$-regular graph $G$, we show that $\left|S\right|=\frac{\beta \left(G\right)+m\left(S\right)}{k?1}$, where $\beta \left(G\right)$ is the cycle rank of $G$, $m\left(S\right)=c+\left|E\left(S\right)\right|?1$ is the margin number of $S$, $c$ and $\left|E\left(S\right)\right|$ are, respectively, the number of components of $G?S$ and the number of edges in $G\left[S\right]$. In particular, for any $?$-set $S$ of a 3-regular graph $G$, we prove that $m\left(S\right)=\xi \left(G\right)$, where $\xi \left(G\right)$ is the Betti deficiency of $G$. This implies that the decycling number of a 3-regular graph $G$ is $\frac{\beta \left(G\right)+\xi \left(G\right)}{2}$. Hence $?\left(G\right)=?\frac{\beta \left(G\right)}{2}?$ for a 3-regular upper-embeddable graph $G$, which concludes the results in [Gao et al., 2015, Wei and Li, 2013] and solves two open problems posed by Bau and Beineke (2002). Considering an algorithm by Furst et al., (1988), there exists a polynomial time algorithm to compute $Z\left(G\right)$, the cardinality of a maximum nonseparating independent set in a $3$-regular graph $G$, which solves an open problem raised by Speckenmeyer (1988). As for a 4-regular graph $G$, we show that for any $?$-set $S$ of $G$, there exists a spanning tree $T$ of $G$ such that the elements of $S$ are simply the leaves of $T$ with at most two exceptions providing $?\left(G\right)=?\frac{\beta \left(G\right)}{3}?$. On the other hand, if $G$ is a loopless graph on $n$ vertices with maximum degree at most $4$, then

The above two upper bounds are tight, and this makes an extension of a result due to Punnim (2006).