A Brooks type theorem for the maximum local edge connectivity

For a graph $G$, let \cn(G) and \la(G) denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \cite{Dirac53} implies that every graph $G$ satisfies \cn(G)\leq \la(G)+1. In this paper we characterize the graphs $G$ for which \cn(G)=\la(G)+1. The case \la(G)=3 was already solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}. We show that a graph $G$ with \la(G)=k\geq 4 satisfies \cn(G)=k+1 if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Haj\'os join.Comment: 15 pages, 1 figur

Extremal Colorings and Independent Sets

We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. It was conjectured that extremal graphs are those which have clique number k and size (k2)+n−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(k2)+n−k(k2)+n−k. We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is Kk∪En−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eKk∪En−kKk∪En−k, K1∨(Kk−1∪En−k) role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eK1∨(Kk−1∪En−k)K1∨(Kk−1∪En−k) and (K1∨(Kk−1∪En−k−c+1))∪Ec−1 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(K1∨(Kk−1∪En−k−c+1))∪Ec−1(K1∨(Kk−1∪En−k−c+1))∪Ec−1 respectively

Graphs whose edge set can be partitioned into maximum matchings

This article provides structural characterization of simple graphs whose edge-set can be partitioned into maximum matchings. We use Vizing's classification of simple graphs based on edge chromatic index

Integer round-up property for the chromatic number of some h-perfect graphs

A graph is h-perfect if its stable set polytope can be completely described by non-negativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph $G$ and every $c\in\mathbb{Z}_{+}^{V(G)}$, the weighted chromatic number of $(G,c)$ is the minimum cardinality of a multi-set $\mathcal{F}$ of stable sets of $G$ such that every $v\in V(G)$ belongs to at least $c_v$ members of $\mathcal{F}$. We prove that every h-perfect line-graph and every t-perfect claw-free graph $G$ has the integer round-up property for the chromatic number: for every non-negative integer weight $c$ on the vertices of $G$, the weighted chromatic number of $(G,c)$ can be obtained by rounding up its fractional relaxation. In other words, the stable set polytope of $G$ has the integer decomposition property. Our results imply the existence of a polynomial-time algorithm which computes the weighted chromatic number of t-perfect claw-free graphs and h-perfect line-graphs. Finally, they yield a new case of a conjecture of Goldberg and Seymour on edge-colorings.Comment: 20 pages, 13 figure

Packing chromatic vertex-critical graphs

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical

Powers of squarefree monomial ideals and combinatorics

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.Comment: 18 pages, 2 figure

Nordhaus-Gaddum Theorem for the Distinguishing Chromatic Number

Nordhaus and Gaddum proved, for any graph G, that the chromatic number of G plus the chromatic number of G complement is less than or equal to the number of vertices in G plus 1. Finck characterized the class of graphs that satisfy equality in this bound. In this paper, we provide a new characterization of this class of graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of the chromatic number of graphs in this class. Our motivation comes from our theorem that generalizes the Nordhaus-Gaddum theorem to the distinguishing chromatic number: for any graph G, the distinguishing chromatic number of G plus the distinguishing chromatic number of G complement is less than or equal to the number of vertices of G plus the distinguishing number of G. Finally, we characterize those graphs that achieve equality in the sum upper bounds simultaneously for both the chromatic number and for our distinguishing chromatic number analog of the Nordhaus-Gaddum inequality.Comment: 18 page

Ear-decompositions and the complexity of the matching polytope

The complexity of the matching polytope of graphs may be measured with the maximum length $\beta$ of a starting sequence of odd ears in an ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its facets are defined by 2-connected factor-critical graphs, which have an odd ear-decomposition (according to a theorem of Lov\'asz). In particular, $\beta(G) \leq 1$ if and only if the matching polytope of the graph $G$ is completely described by non-negativity, star and odd-circuit inequalities. This is essentially equivalent to the h-perfection of the line-graph of $G$, as observed by Cao and Nemhauser. The complexity of computing $\beta$ is apparently not known. We show that deciding whether $\beta(G)\leq 1$ can be executed efficiently by looking at any ear-decomposition starting with an odd circuit and performing basic modulo-2 computations. Such a greedy-approach is surprising in view of the complexity of the problem in more special cases by Bruhn and Schaudt, and it is simpler than using the Parity Minor Algorithm. Our results imply a simple polynomial-time algorithm testing h-perfection in line-graphs (deciding h-perfection is open in general). We also generalize our approach to binary matroids and show that computing $\beta$ is a Fixed-Parameter-Tractable problem (FPT)

Structural Properties of Index Coding Capacity Using Fractional Graph Theory

The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on Information Theory (ISIT

Some concepts in list coloring

In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k-list colorable if it admits a k-list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k-list colorable is called the M-number of G. We show that every triangle-free uniquely vertex colorable graph with chromatic number k+1, is uniquely k-list colorable. A bound for the M-number of graphs is given, and using this bound it is shown that every planar graph has M-number at most 4. Also we introduce list criticality in graphs and characterize all 3-list critical graphs. It is conjectured that every $\chi_\ell$-critical graph is $\chi'$-critical and the equivalence of this conjecture to the well known list coloring conjecture is shown